{"id":1239,"date":"2026-11-09T18:18:33","date_gmt":"2026-11-09T18:18:33","guid":{"rendered":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/?p=1239"},"modified":"2026-11-10T06:34:37","modified_gmt":"2026-11-10T06:34:37","slug":"the-world-of-algorithms","status":"publish","type":"post","link":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/2026\/11\/09\/the-world-of-algorithms\/","title":{"rendered":"\u0905\u0927\u094d\u092f\u093e\u092f 11 &#8220;\u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u0915\u0940 \u0926\u0941\u0928\u093f\u092f\u093e&#8221;"},"content":{"rendered":"<div id=\"pl-1239\"  class=\"panel-layout\" ><div id=\"pg-1239-0\"  class=\"panel-grid panel-no-style\" ><div id=\"pgc-1239-0-0\"  class=\"panel-grid-cell\" ><div id=\"panel-1239-0-0-0\" class=\"widget_text so-panel widget widget_custom_html panel-first-child panel-last-child\" data-index=\"0\" ><div class=\"textwidget custom-html-widget\"><div class=\"pargai-multislider\">\n    <div class=\"slider-header-section\">\n        <div class=\"header-main-title\">\u092d\u093e\u0917 1: \u0905\u0927\u094d\u092f\u093e\u092f 11 - \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u0915\u0940 \u0926\u0941\u0928\u093f\u092f\u093e (\u092a\u0930\u093f\u091a\u092f)<\/div>\n    <\/div>\n    \n    <div class=\"slider-content-area\">\n        <div class=\"pargai-slide active-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 1: \u0905\u0927\u094d\u092f\u093e\u092f \u0915\u0940 \u092a\u094d\u0930\u0938\u094d\u0924\u093e\u0935\u0928\u093e:<\/b><\/p>\n            <p>\u0906\u091c\u0915\u0932 '\u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e' \u0936\u092c\u094d\u0926 \u0939\u0930 \u091c\u0917\u0939 \u0938\u0941\u0928\u093e\u0908 \u0926\u0947\u0924\u093e \u0939\u0948\u0964 \u0907\u0938 \u0905\u0927\u094d\u092f\u093e\u092f \u092e\u0947\u0902 \u0939\u092e \u0938\u092e\u091d\u0947\u0902\u0917\u0947 \u0915\u093f \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u0915\u094d\u092f\u093e \u0939\u0948 \u0914\u0930 \u0907\u0938\u0915\u093e \u0917\u0923\u093f\u0924 \u0938\u0947 \u0915\u094d\u092f\u093e \u0938\u0902\u092c\u0902\u0927 \u0939\u0948[cite: 1, 2]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 2: \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u094b \u0917\u093f\u0928\u0928\u0947 \u0915\u0940 \u092a\u094d\u0930\u093e\u091a\u0940\u0928 \u0935\u093f\u0927\u093f:<\/b><\/p>\n            <p>\u0936\u0941\u0930\u0941\u0906\u0924 \u092e\u0947\u0902 \u0939\u092e \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u094b \u092c\u093f\u0902\u0926\u0941 (dots) \u0916\u0940\u0902\u091a\u0915\u0930 \u0914\u0930 \u0909\u0928\u094d\u0939\u0947\u0902 \u0917\u093f\u0928\u0915\u0930 \u091c\u094b\u0921\u093c\u0928\u093e \u0938\u0940\u0916\u0924\u0947 \u0925\u0947, \u091c\u0948\u0938\u0947 5 \u0914\u0930 7 \u0915\u0947 \u0932\u093f\u090f 5 \u0914\u0930 7 \u092c\u093f\u0902\u0926\u0941 \u092c\u0928\u093e\u0928\u093e[cite: 1]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 3: \u092c\u0921\u093c\u0940 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0940 \u0938\u092e\u0938\u094d\u092f\u093e:<\/b><\/p>\n            <p>\u092f\u0926\u093f \u092c\u0921\u093c\u0940 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u091c\u0948\u0938\u0947 473 + 695 \u0915\u094b \u0907\u0938 \u0924\u0930\u0939 \u092c\u093f\u0902\u0926\u0941\u0913\u0902 \u0938\u0947 \u091c\u094b\u0921\u093c\u093e \u091c\u093e\u090f, \u0924\u094b \u092f\u0939 \u092c\u0947\u0939\u0926 \u0915\u0920\u093f\u0928 \u0914\u0930 \u0938\u092e\u092f \u0932\u0947\u0928\u0947 \u0935\u093e\u0932\u093e \u0939\u094b \u091c\u093e\u0924\u093e \u0939\u0948[cite: 1]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 4: \u092d\u093e\u0930\u0924\u0940\u092f \u0938\u0902\u0916\u094d\u092f\u093e \u092a\u094d\u0930\u0923\u093e\u0932\u0940 \u0915\u093e \u0935\u093f\u0915\u093e\u0938:<\/b><\/p>\n            <p>\u0939\u092e\u0928\u0947 \u090f\u0915 \u0924\u0947\u091c \u0924\u0930\u0940\u0915\u093e \u0916\u094b\u091c\u093e \u091c\u093f\u0938\u092e\u0947\u0902 \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u094b \u092d\u093e\u0930\u0924\u0940\u092f \u0906\u0927\u093e\u0930-\u0926\u0938 \u0938\u094d\u0925\u093e\u0928\u0940\u092f \u092e\u093e\u0928 \u092a\u094d\u0930\u0923\u093e\u0932\u0940 \u092e\u0947\u0902 \u0926\u093e\u090f\u0902 \u0938\u0947 \u092c\u093e\u090f\u0902 \u0938\u0902\u0930\u0947\u0916\u093f\u0924 \u0915\u0930\u0915\u0947 \u091c\u094b\u0921\u093c\u093e \u091c\u093e\u0924\u093e \u0939\u0948[cite: 2]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 5: \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e:<\/b><\/p>\n            <p>\u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u0915\u093f\u0938\u0940 \u0909\u0924\u094d\u0924\u0930 \u0924\u0915 \u092a\u0939\u0941\u0901\u091a\u0928\u0947 \u0915\u0940 \u090f\u0915 \u091a\u0930\u0923-\u0926\u0930-\u091a\u0930\u0923 (Step-by-step) \u0935\u094d\u092f\u0935\u0938\u094d\u0925\u093f\u0924 \u0914\u0930 \u0938\u091f\u0940\u0915 \u092a\u094d\u0930\u0915\u094d\u0930\u093f\u092f\u093e \u0939\u0948[cite: 2]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 6: \u0938\u092e\u092f \u0914\u0930 \u0926\u0915\u094d\u0937\u0924\u093e \u0915\u0940 \u0924\u0941\u0932\u0928\u093e:<\/b><\/p>\n            <p>\u092c\u093f\u0902\u0926\u0941\u0913\u0902 \u0915\u094b \u0917\u093f\u0928\u0928\u0947 \u0915\u0940 \u0924\u0941\u0932\u0928\u093e \u092e\u0947\u0902 \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u0906\u0927\u093e\u0930\u093f\u0924 \u0917\u0923\u0928\u093e \u0928 \u0915\u0947\u0935\u0932 \u0924\u0947\u091c \u0939\u094b\u0924\u0940 \u0939\u0948, \u092c\u0932\u094d\u0915\u093f \u0936\u094d\u0930\u092e \u0914\u0930 \u0938\u092e\u092f \u0915\u0940 \u092d\u093e\u0930\u0940 \u092c\u091a\u0924 \u0915\u0930\u0924\u0940 \u0939\u0948[cite: 3]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 7: \u092e\u0939\u0924\u094d\u0924\u092e \u0938\u092e\u093e\u092a\u0935\u0930\u094d\u0924\u094d\u092f (GCD) \u0915\u093e \u092a\u0930\u093f\u091a\u092f:<\/b><\/p>\n            <p>\u0905\u0927\u094d\u092f\u093e\u092f \u0915\u0947 \u0926\u0942\u0938\u0930\u0947 \u092d\u093e\u0917 \u092e\u0947\u0902 \u0939\u092e \u0926\u094b \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u092e\u0939\u0924\u094d\u0924\u092e \u0938\u092e\u093e\u092a\u0935\u0930\u094d\u0924\u0915 \u092f\u093e \u0909\u091a\u094d\u091a\u0924\u092e \u0909\u092d\u092f\u0928\u093f\u0937\u094d\u0920 \u0917\u0941\u0923\u0928\u0916\u0902\u0921 (HCF\/GCD) \u0928\u093f\u0915\u093e\u0932\u0928\u0947 \u0915\u0947 \u0924\u0930\u0940\u0915\u094b\u0902 \u0915\u093e \u0905\u0927\u094d\u092f\u092f\u0928 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902[cite: 4]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 8: \u0921\u0947\u091f\u093e \u0938\u094d\u091f\u094d\u0930\u0915\u094d\u091a\u0930 \u0914\u0930 \u0938\u0942\u091a\u093f\u092f\u093e\u0901:<\/b><\/p>\n            <p>\u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u092e\u0947\u0902 \u092e\u0927\u094d\u092f\u0935\u0930\u094d\u0924\u0940 \u092e\u093e\u0928\u094b\u0902 \u0915\u094b \u0938\u0941\u0930\u0915\u094d\u0937\u093f\u0924 \u0930\u0916\u0928\u0947 \u0914\u0930 \u0935\u094d\u092f\u0935\u0938\u094d\u0925\u093f\u0924 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f '\u0938\u0942\u091a\u093f\u092f\u094b\u0902' (Lists) \u091c\u0948\u0938\u0940 \u0921\u0947\u091f\u093e \u0938\u0902\u0930\u091a\u0928\u093e\u0913\u0902 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948[cite: 8]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 9: \u092f\u0942\u0915\u094d\u0932\u093f\u0921 \u0914\u0930 \u0906\u0930\u094d\u092f\u092d\u091f\u094d\u091f \u0915\u0947 \u0924\u0930\u0940\u0915\u0947:<\/b><\/p>\n            <p>\u0939\u092e GCD \u0928\u093f\u0915\u093e\u0932\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u092f\u0942\u0915\u094d\u0932\u093f\u0921 \u0915\u0947 \u0918\u091f\u093e\u0935 \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u0914\u0930 \u0906\u0930\u094d\u092f\u092d\u091f\u094d\u091f \u0915\u0940 \u0909\u0928\u094d\u0928\u0924 \u0932\u0902\u092c\u0940 \u092d\u093e\u0917 \u0935\u093f\u0927\u093f (Division Algorithm) \u0915\u094b \u0938\u0940\u0916\u0924\u0947 \u0939\u0948\u0902[cite: 11, 13]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 10: \u0905\u0927\u094d\u092f\u093e\u092f \u0915\u093e \u092e\u0941\u0916\u094d\u092f \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f:<\/b><\/p>\n            <p>\u0938\u092e\u0938\u094d\u092f\u093e\u0913\u0902 \u0915\u094b \u0939\u0932 \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0924\u093e\u0930\u094d\u0915\u093f\u0915, \u0938\u094d\u092a\u0937\u094d\u091f \u0914\u0930 \u091a\u0930\u0923\u092c\u0926\u094d\u0927 \u092a\u094d\u0930\u0915\u094d\u0930\u093f\u092f\u093e\u090f\u0901 \u0935\u093f\u0915\u0938\u093f\u0924 \u0915\u0930\u0928\u093e \u0914\u0930 \u0909\u0928\u0915\u0940 \u0926\u0915\u094d\u0937\u0924\u093e \u0915\u093e \u0935\u093f\u0936\u094d\u0932\u0947\u0937\u0923 \u0915\u0930\u0928\u093e \u0907\u0938 \u0905\u0927\u094d\u092f\u093e\u092f \u0915\u093e \u092e\u0941\u0916\u094d\u092f \u0909\u0926\u094d\u0926\u0947\u0936\u094d\u092f \u0939\u0948\u3002<\/p>\n        <\/div>\n    <\/div>\n\n    <div class=\"slider-footer-section\">\n        <button class=\"slider-nav-btn btn-prev\">\u092a\u093f\u091b\u0932\u093e (Prev)<\/button>\n        <span class=\"slider-page-counter\">1 \/ 10<\/span>\n        <button class=\"slider-nav-btn btn-next\">\u0905\u0917\u0932\u093e (Next)<\/button>\n    <\/div>\n<\/div><\/div><\/div><\/div><\/div><div id=\"pg-1239-1\"  class=\"panel-grid panel-no-style\" ><div id=\"pgc-1239-1-0\"  class=\"panel-grid-cell\" ><div id=\"panel-1239-1-0-0\" class=\"widget_text so-panel widget widget_custom_html panel-first-child panel-last-child\" data-index=\"1\" ><div class=\"textwidget custom-html-widget\"><div class=\"pargai-multislider\">\n    <div class=\"slider-header-section\">\n        <div class=\"header-main-title\">\u092d\u093e\u0917 2: \u0905\u0927\u094d\u092f\u093e\u092f 11 - \u092e\u0941\u0916\u094d\u092f \u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924 \u0914\u0930 \u0928\u093f\u092f\u092e<\/div>\n    <\/div>\n    \n    <div class=\"slider-content-area\">\n        <div class=\"pargai-slide active-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 1: \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u094b \u091c\u094b\u0921\u093c\u0928\u0947 \u0915\u093e \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e:<\/b><\/p>\n            <p>\u091a\u0930\u0923 1: \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u094b \u0926\u093e\u090f\u0902 \u0938\u0947 \u092c\u093e\u090f\u0902 \u0938\u0902\u0930\u0947\u0916\u093f\u0924 \u0915\u0930\u0947\u0902\u0964 \u091a\u0930\u0923 2: \u0938\u092c\u0938\u0947 \u0926\u093e\u090f\u0902 \u0905\u0902\u0915\u094b\u0902 \u0915\u094b \u091c\u094b\u0921\u093c\u0947\u0902; \u092f\u0926\u093f \u092f\u094b\u0917 10 \u0938\u0947 \u0905\u0927\u093f\u0915 \u0939\u094b \u0924\u094b \u0907\u0915\u093e\u0908 \u0928\u0940\u091a\u0947 \u0932\u093f\u0916\u0947\u0902 \u0914\u0930 \u0939\u093e\u0938\u093f\u0932 1 \u0938\u0947\u091f \u0915\u0930\u0947\u0902[cite: 2, 3]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 2: \u0939\u093e\u0938\u093f\u0932 (Carry) \u0915\u093e \u0928\u093f\u092f\u092e:<\/b><\/p>\n            <p>\u092f\u094b\u0917 \u0915\u0940 \u092a\u094d\u0930\u0915\u094d\u0930\u093f\u092f\u093e \u092e\u0947\u0902 \u0915\u093f\u0938\u0940 \u092d\u0940 \u0915\u0949\u0932\u092e \u092e\u0947\u0902 \u0939\u093e\u0938\u093f\u0932 \u0915\u093e \u092e\u093e\u0928 \u0915\u092d\u0940 \u092d\u0940 1 \u0938\u0947 \u0905\u0927\u093f\u0915 \u0928\u0939\u0940\u0902 \u0939\u094b \u0938\u0915\u0924\u093e, \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0905\u0927\u093f\u0915\u0924\u092e \u0938\u0902\u092d\u0935 \u092f\u094b\u0917 19 \u0939\u094b\u0924\u093e \u0939\u0948[cite: 3]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 3: \u0935\u093f\u092d\u093e\u091c\u0915 (Divisors) \u0916\u094b\u091c\u0928\u0947 \u0915\u093e \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e:<\/b><\/p>\n            <p>\u0915\u093f\u0938\u0940 \u0938\u0902\u0916\u094d\u092f\u093e n \u0915\u0947 \u0935\u093f\u092d\u093e\u091c\u0915 \u0922\u0942\u0901\u0922\u0646\u06d2 \u0915\u0947 \u0932\u093f\u090f 1 \u0938\u0947 n \u0924\u0915 \u0915\u0940 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0938\u0902\u0916\u094d\u092f\u093e j \u0915\u0940 \u091c\u093e\u0901\u091a \u0915\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948; \u092f\u0926\u093f j, n \u0915\u094b \u0935\u093f\u092d\u093e\u091c\u093f\u0924 \u0915\u0930\u0947, \u0924\u094b \u0909\u0938\u0947 \u0938\u0942\u091a\u0940 \u092e\u0947\u0902 \u091c\u094b\u0921\u093c\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 4: \u0909\u092d\u092f\u0928\u093f\u0937\u094d\u0920 \u0935\u093f\u092d\u093e\u091c\u0915 (Common Divisors):<\/b><\/p>\n            <p>\u0926\u094b \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 m \u0914\u0930 n \u0915\u0940 \u0935\u093f\u092d\u093e\u091c\u0915 \u0938\u0942\u091a\u093f\u092f\u094b\u0902 \u0915\u0940 \u0924\u0941\u0932\u0928\u093e \u0915\u0930\u0915\u0947 \u0909\u092d\u092f\u0928\u093f\u0937\u094d\u0920 \u0935\u093f\u092d\u093e\u091c\u0915\u094b\u0902 \u0915\u0940 \u0938\u0942\u091a\u0940 \u092c\u0928\u093e\u0908 \u091c\u093e\u0924\u0940 \u0939\u0948, \u091c\u093f\u0938\u0915\u093e \u0938\u092c\u0938\u0947 \u0926\u093e\u0939\u093f\u0928\u093e \u0924\u0924\u094d\u0935 GCD \u0939\u094b\u0924\u093e \u0939\u0948\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 5: \u092f\u0942\u0915\u094d\u0932\u093f\u0921 \u0915\u093e \u0918\u091f\u093e\u0935 \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e:<\/b><\/p>\n            <p>\u092f\u0926\u093f m \u2265 n \u0939\u0948, \u0924\u094b \u092f\u0942\u0915\u094d\u0932\u093f\u0921 \u0915\u0947 \u0928\u093f\u092f\u092e\u0928\u0941\u0938\u093e\u0930: $\\text{gcd}(m, n) = \\text{gcd}(n, m - n)$ \u0939\u094b\u0924\u093e \u0939\u0948[cite: 11]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 6: \u0906\u0930\u094d\u092f\u092d\u091f\u094d\u091f \u0915\u0940 \u0935\u093f\u092d\u093e\u091c\u0928 \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e:<\/b><\/p>\n            <p>\u0918\u091f\u093e\u0928\u0947 \u0915\u0940 \u092a\u094d\u0930\u0915\u094d\u0930\u093f\u092f\u093e \u0915\u094b \u0924\u0947\u091c \u0915\u0930\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u0936\u0947\u0937\u092b\u0932 \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u093f\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948: $\\text{gcd}(m, n) = \\text{gcd}(n, m \\pmod n)$[cite: 13]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 7: \u0905\u092d\u093e\u091c\u094d\u092f \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u0940 \u092a\u0930\u093f\u092d\u093e\u0937\u093e:<\/b><\/p>\n            <p>\u090f\u0915 \u0905\u092d\u093e\u091c\u094d\u092f \u0938\u0902\u0916\u094d\u092f\u093e \u0935\u0939 \u0939\u094b\u0924\u0940 \u0939\u0948 \u091c\u093f\u0938\u0915\u0947 \u0920\u0940\u0915 \u0926\u094b \u092d\u093f\u0928\u094d\u0928 \u0917\u0941\u0923\u0928\u0916\u0902\u0921 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u20141 \u0914\u0930 \u0935\u0939 \u0938\u0902\u0916\u094d\u092f\u093e \u0938\u094d\u0935\u092f\u0902\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 8: \u0905\u092d\u093e\u091c\u094d\u092f \u0917\u0941\u0923\u0928\u0916\u0902\u0921\u0928 (Prime Factorization):<\/b><\/p>\n            <p>\u0915\u093f\u0938\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u094b \u0905\u092d\u093e\u091c\u094d\u092f \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0947 \u0918\u093e\u0924\u094b\u0902 \u0915\u0947 \u0917\u0941\u0923\u0928\u092b\u0932 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0935\u094d\u092f\u0915\u094d\u0924 \u0915\u0930\u0928\u093e, \u091c\u0948\u0938\u0947 $180 = 2^2 \\times 3^2 \\times 5^1$[cite: 14]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 9: \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u0915\u0940 \u0926\u0915\u094d\u0937\u0924\u093e (Efficiency):<\/b><\/p>\n            <p>\u092c\u0947\u0939\u0924\u0930 \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u0935\u0947 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902 \u091c\u093f\u0928\u0915\u093e \u092a\u094d\u0930\u092f\u093e\u0938 (Effort) \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u0947 \u092e\u093e\u0928 \u0915\u0947 \u092c\u091c\u093e\u092f \u0909\u0938\u0915\u0947 \u0905\u0902\u0915\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u0947 \u0938\u092e\u093e\u0928\u0941\u092a\u093e\u0924\u0940 \u0939\u094b\u0924\u093e \u0939\u0948[cite: 14]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 10: \u0910\u0924\u093f\u0939\u093e\u0938\u093f\u0915 \u092a\u0943\u0937\u094d\u0920\u092d\u0942\u092e\u093f (\u0905\u0932-\u0916\u094d\u0935\u093e\u0930\u093f\u091c\u094d\u092e\u0940):<\/b><\/p>\n            <p>\u092b\u093e\u0930\u0938\u0940 \u0917\u0923\u093f\u0924\u091c\u094d\u091e \u0905\u0932-\u0916\u094d\u0935\u093e\u0930\u093f\u091c\u094d\u092e\u0940 \u0928\u0947 \u092d\u093e\u0930\u0924\u0940\u092f \u0905\u0902\u0915\u0917\u0923\u093f\u0924 \u0914\u0930 \u0938\u094d\u0925\u093e\u0928-\u092e\u093e\u0928 \u092a\u094d\u0930\u0923\u093e\u0932\u0940 \u092a\u0930 \u0917\u094d\u0930\u0902\u0925 \u0932\u093f\u0916\u093e, \u091c\u093f\u0938\u0915\u0947 \u0932\u0948\u091f\u093f\u0928 \u0905\u0928\u0941\u0935\u093e\u0926 \u0938\u0947 'Algorithm' \u0936\u092c\u094d\u0926 \u092c\u0928\u093e[cite: 14]\u3002<\/p>\n        <\/div>\n    <\/div>\n\n    <div class=\"slider-footer-section\">\n        <button class=\"slider-nav-btn btn-prev\">\u092a\u093f\u091b\u0932\u093e (Prev)<\/button>\n        <span class=\"slider-page-counter\">1 \/ 10<\/span>\n        <button class=\"slider-nav-btn btn-next\">\u0905\u0917\u0932\u093e (Next)<\/button>\n    <\/div>\n<\/div><\/div><\/div><\/div><\/div><div id=\"pg-1239-2\"  class=\"panel-grid panel-no-style\" ><div id=\"pgc-1239-2-0\"  class=\"panel-grid-cell\" ><div id=\"panel-1239-2-0-0\" class=\"widget_text so-panel widget widget_custom_html panel-first-child panel-last-child\" data-index=\"2\" ><div class=\"textwidget custom-html-widget\"><!-- ================= \u092c\u094d\u0932\u0949\u0915 3: \u0905\u092d\u094d\u092f\u093e\u0938 \u0914\u0930 \u0935\u093f\u0938\u094d\u0924\u0943\u0924 \u0939\u0932 (Exercise Solutions) ================= -->\n<div class=\"pargai-multislider\">\n    <div class=\"slider-header-section\">\n        <div class=\"header-main-title\">\u092d\u093e\u0917 3: \u0905\u0927\u094d\u092f\u093e\u092f 11 - \u0905\u092d\u094d\u092f\u093e\u0938 \u0914\u0930 \u0935\u093f\u0938\u094d\u0924\u0943\u0924 \u0939\u0932 (Exercise Solutions)<\/div>\n    <\/div>\n    \n    <div class=\"slider-content-area\">\n        <div class=\"pargai-slide active-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 1: Exercise Set 11.1 (\u092a\u094d\u0930\u0936\u094d\u0928 1 & 2):<\/b><\/p>\n            <p><b>\u092a\u094d\u0930\u0936\u094d\u0928 1:<\/b> \u0926\u094b 4-\u0905\u0902\u0915\u0940\u092f \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u094b \u091a\u0930\u0923\u094b\u0902 \u0915\u0947 \u0905\u0928\u0941\u0938\u093e\u0930 \u091c\u094b\u0921\u093c\u0947\u0902 \u0914\u0930 \u0938\u0939\u0940 \u092a\u0930\u093f\u0923\u093e\u092e \u092a\u094d\u0930\u093e\u092a\u094d\u0924 \u0915\u0930\u0947\u0902[cite: 19]\u0964 <br><b>\u0939\u0932:<\/b> \u0926\u093e\u090f\u0902 \u0938\u0947 \u092c\u093e\u090f\u0902 \u0915\u0949\u0932\u092e \u091c\u094b\u0921\u093c\u0928\u0947 \u092a\u0930 (\u091c\u0948\u0938\u0947 4734 + 2695 = 7429) \u0938\u0939\u0940 \u0909\u0924\u094d\u0924\u0930 \u092e\u093f\u0932\u0924\u093e \u0939\u0948[cite: 1, 3]\u0964 <br><b>\u092a\u094d\u0930\u0936\u094d\u0928 2:<\/b> 5-\u0905\u0902\u0915\u0940\u092f \u0914\u0930 3-\u0905\u0902\u0915\u0940\u092f \u0938\u0902\u0916\u094d\u092f\u093e \u091c\u094b\u0921\u093c\u0928\u0947 \u092a\u0930 \u091a\u0930\u0923 \u0938\u0939\u0940 \u0922\u0902\u0917 \u0938\u0947 \u0915\u093e\u092e \u0915\u0930\u0924\u0947 \u0939\u0948\u0902, \u0916\u093e\u0932\u0940 \u0938\u094d\u0925\u093e\u0928\u094b\u0902 \u092a\u0930 '0' \u092e\u093e\u0928\u093e \u091c\u093e\u0924\u093e \u0939\u0948[cite: 19]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 2: Exercise Set 11.1 (\u092a\u094d\u0930\u0936\u094d\u0928 3, 4 & 5):<\/b><\/p>\n            <p><b>\u092a\u094d\u0930\u0936\u094d\u0928 3:<\/b> \u0915\u0949\u0932\u092e \u0915\u094b \u0926\u093e\u090f\u0902 \u0938\u0947 \u092c\u093e\u090f\u0902 \u0915\u094d\u092f\u094b\u0902 \u0938\u0902\u0930\u0947\u0916\u093f\u0924 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902? \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0917\u0923\u0928\u093e \u0907\u0915\u093e\u0908 \u0938\u094d\u0925\u093e\u0928 \u0938\u0947 \u0936\u0941\u0930\u0942 \u0939\u094b\u0924\u0940 \u0939\u0948\u0964 <br><b>\u092a\u094d\u0930\u0936\u094d\u0928 4:<\/b> \u091a\u0930\u0923 3 \u092e\u0947\u0902 \u0939\u093e\u0938\u093f\u0932 1 \u0938\u0947 \u0905\u0927\u093f\u0915 \u0915\u094d\u092f\u094b\u0902 \u0928\u0939\u0940\u0902 \u0939\u094b \u0938\u0915\u0924\u0940? \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0905\u0927\u093f\u0915\u0924\u092e \u092f\u094b\u0917 9 + 9 + 1 = 19 \u0939\u094b\u0924\u093e \u0939\u0948[cite: 3]\u0964 <br><b>\u092a\u094d\u0930\u0936\u094d\u0928 5:<\/b> \u092a\u093e\u0901\u091a\u0935\u0947\u0902 \u091a\u0930\u0923 (\u0905\u0902\u0924\u093f\u092e \u0915\u0948\u0930\u0940) \u0915\u0947 \u092c\u093f\u0928\u093e \u092c\u0921\u093c\u0947 \u092f\u094b\u0917 (\u091c\u0948\u0938\u0947 75 + 50) \u0935\u093f\u092b\u0932 \u0939\u094b \u0938\u0915\u0924\u0947 \u0939\u0948\u0902[cite: 19]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 3: Exercise Set 11.2 (\u092a\u094d\u0930\u0936\u094d\u0928 1 & 2):<\/b><\/p>\n            <p><b>\u092a\u094d\u0930\u0936\u094d\u0928 1:<\/b> \u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f List 1 \u0914\u0930 List 2 \u092c\u0922\u093c\u0924\u0940 \u0939\u0941\u0908 \u0915\u094d\u0930\u092e \u092e\u0947\u0902 \u0939\u0948\u0902\u0964 (i) List 1 \u0915\u0947 \u0935\u0947 \u0924\u0924\u094d\u0935 \u0922\u0942\u0901\u0922\u0947\u0902 \u091c\u094b List 2 \u092e\u0947\u0902 \u0928\u0939\u0940\u0902 \u0939\u0948\u0902[cite: 24]\u0964 (ii) List 2 \u0915\u0947 \u0935\u0947 \u0924\u0924\u094d\u0935 \u0922\u0942\u0901\u0922\u0947\u0902 \u091c\u094b List 1 \u092e\u0947\u0902 \u0928\u0939\u0940\u0902 \u0939\u0948\u0902[cite: 24]\u0964<br><b>\u0939\u0932:<\/b> \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0924\u0924\u094d\u0935 \u0915\u0940 \u0926\u0942\u0938\u0930\u0947 \u0938\u0942\u091a\u0940 \u092e\u0947\u0902 \u0909\u092a\u0938\u094d\u0925\u093f\u0924\u093f \u0915\u0940 \u091c\u093e\u0901\u091a \u0915\u0930\u0915\u0947 \u0928\u0908 \u0938\u0942\u091a\u0940 \u092c\u0928\u093e\u0908 \u091c\u093e\u0924\u0940 \u0939\u0948\u0964 <br><b>\u092a\u094d\u0930\u0936\u094d\u0928 2:<\/b> LCM \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u093e \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e: lcm(m, n) = (m &times; n) \/ gcd(m, n)[cite: 24]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 4: Exercise Set 11.2 (\u092a\u094d\u0930\u0936\u094d\u0928 3 - Divisor Pairs):<\/b><\/p>\n            <p><b>\u092a\u094d\u0930\u0936\u094d\u0928 3:<\/b> \u0935\u093f\u092d\u093e\u091c\u0915 \u091c\u094b\u0921\u093c\u094b\u0902 \u092e\u0947\u0902 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902 (\u091c\u0948\u0938\u0947 18 \u0915\u0947 \u091c\u094b\u0921\u093c\u0947: (1,18), (2,9), (3,6))[cite: 25]\u0964 <br>(i) \u091c\u094b\u0921\u093c\u0947 \u092e\u0947\u0902 \u0932\u093f\u0916\u0928\u0947 \u092a\u0930 1 \u0938\u0947 n \u0924\u0915 \u092e\u0947\u0902 \u0915\u0947\u0935\u0932 &radic;n (\u0935\u0930\u094d\u0917\u092e\u0942\u0932 n) \u0924\u0915 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u0926\u0947\u0916\u0928\u0940 \u092a\u0921\u093c\u0924\u0940 \u0939\u0948\u0902[cite: 25]\u0964 <br>(ii) \u0915\u094d\u092f\u093e GCD \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u0935\u0948\u0938\u0947 \u0939\u0940 \u0915\u093e\u092e \u0915\u0930\u0947\u0917\u093e? \u0939\u093e\u0901, \u0935\u094d\u092f\u0935\u0938\u094d\u0925\u093f\u0924 \u091c\u093e\u0901\u091a \u0938\u0947 \u092a\u0930\u093f\u0923\u093e\u092e \u0938\u092e\u093e\u0928 \u0930\u0939\u0924\u093e \u0939\u0948[cite: 25]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 5: Exercise Set 11.3 (\u092a\u094d\u0930\u0936\u094d\u0928 1 & 2):<\/b><\/p>\n            <p><b>\u092a\u094d\u0930\u0936\u094d\u0928 1:<\/b> \u092f\u0926\u093f divisors(n) \u0915\u0940 \u0917\u0923\u0928\u093e n \u0938\u0947 1 \u0915\u0940 \u0913\u0930 (reverse order) \u0915\u0940 \u091c\u093e\u090f, \u0924\u094b \u0938\u0942\u091a\u0940 \u0918\u091f\u0924\u0947 \u0915\u094d\u0930\u092e \u0915\u0947 \u092c\u091c\u093e\u092f \u0905\u0932\u0917 \u0915\u094d\u0930\u092e \u092e\u0947\u0902 \u092c\u0928\u0947\u0917\u0940[cite: 26]\u0964 <br><b>\u092a\u094d\u0930\u0936\u094d\u0928 2:<\/b> \u092f\u0926\u093f \u0938\u092c\u0938\u0947 \u092c\u0921\u093c\u0947 \u0909\u092d\u092f\u0928\u093f\u0937\u094d\u0920 \u0935\u093f\u092d\u093e\u091c\u0915 \u0938\u0947 \u0936\u0941\u0930\u0942 \u0915\u0930\u0915\u0947 \u092a\u0940\u091b\u0947 (min(m, n) \u0938\u0947 1 \u0915\u0940 \u0913\u0930) \u0915\u093e\u092e \u0915\u0930\u0947\u0902, \u0924\u094b \u0939\u092e\u0947\u0902 \u0938\u092c\u0938\u0947 \u092a\u0939\u0932\u093e \u092e\u093f\u0932\u0928\u0947 \u0935\u093e\u0932\u093e \u0909\u092d\u092f\u0928\u093f\u0937\u094d\u0920 \u0935\u093f\u092d\u093e\u091c\u0915 \u0939\u0940 \u0938\u0940\u0927\u093e GCD \u092e\u093f\u0932 \u091c\u093e\u090f\u0917\u093e[cite: 26]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 6: End-of-Chapter Exercises (\u092a\u094d\u0930\u0936\u094d\u0928 1 - GCD \u0917\u0923\u0928\u093e):<\/b><\/p>\n            <p>\u0906\u0930\u094d\u092f\u092d\u091f\u094d\u091f \u0915\u0940 \u0909\u0928\u094d\u0928\u0924 \u0935\u093f\u092d\u093e\u091c\u0928 \u0935\u093f\u0927\u093f \u0938\u0947 \u0917\u0923\u0928\u093e:<br>\n            (i) gcd(375, 825) = 75<br>\n            (ii) gcd(54000, 81000) = 27000[cite: 6]\n            (iii) gcd(1789287, 237656) = 1<br>\n            (iv) gcd(2587392, 157656) = 24<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 7: End-of-Chapter Exercises (\u092a\u094d\u0930\u0936\u094d\u0928 2):<\/b><\/p>\n            <p><b>\u092a\u094d\u0930\u0936\u094d\u0928 2:<\/b> \u0938\u0924\u094d\u092f\u093e\u092a\u093f\u0924 \u0915\u0930\u0947\u0902 \u0915\u093f m &ge; n \u0915\u0947 \u0932\u093f\u090f, d, m \u0914\u0930 n \u0926\u094b\u0928\u094b\u0902 \u0915\u094b \u0935\u093f\u092d\u093e\u091c\u093f\u0924 \u0915\u0930\u0924\u093e \u0939\u0948 \u092f\u0926\u093f \u0914\u0930 \u0915\u0947\u0935\u0932 \u092f\u0926\u093f d, n \u0914\u0930 (m mod n) \u0926\u094b\u0928\u094b\u0902 \u0915\u094b \u0935\u093f\u092d\u093e\u091c\u093f\u0924 \u0915\u0930\u0924\u093e \u0939\u0948\u3002<br><b>\u0939\u0932:<\/b> \u091f\u093e\u0907\u0932\u093f\u0902\u0917 \u0914\u0930 \u0936\u0947\u0937\u092b\u0932 \u0915\u0947 \u091c\u094d\u092f\u093e\u092e\u093f\u0924\u0940\u092f \u092a\u094d\u0930\u092e\u093e\u0923 \u0938\u0947 \u092f\u0939 \u0938\u093f\u0926\u094d\u0927 \u0939\u094b\u0924\u093e \u0939\u0948 \u0915\u093f \u0926\u094b\u0928\u094b\u0902 \u0915\u0947 \u0938\u093e\u0930\u094d\u0935\u092d\u093e\u091c\u0915 (common divisors) \u0938\u092e\u093e\u0928 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902[cite: 29]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 8: End-of-Chapter Exercises (\u092a\u094d\u0930\u0936\u094d\u0928 3 & 4):<\/b><\/p>\n            <p><b>\u092a\u094d\u0930\u0936\u094d\u0928 3 (prime(n)):<\/b> \u091c\u093e\u0901\u091a \u0915\u0930\u0947\u0902 \u0915\u093f \u0915\u094d\u092f\u093e n \u0905\u092d\u093e\u091c\u094d\u092f \u0939\u0948\u0964 \u092f\u0926\u093f divisors(n) \u0915\u0940 \u0932\u0902\u092c\u093e\u0908 \u0920\u0940\u0915 2 \u0939\u0948 (\u0905\u0930\u094d\u0925\u093e\u0924 [1, n]), \u0924\u094b \u0938\u0902\u0916\u094d\u092f\u093e \u0905\u092d\u093e\u091c\u094d\u092f \u0939\u0948\u3002<br><b>\u092a\u094d\u0930\u0936\u094d\u0928 4 (primedivisors(n)):<\/b> divisors(n) \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0930\u0947\u0902 \u0914\u0930 \u092b\u093f\u0930 \u0909\u0938\u092e\u0947\u0902 \u0938\u0947 \u0915\u0947\u0935\u0932 \u0905\u092d\u093e\u091c\u094d\u092f \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u094b \u091b\u093e\u0901\u091f\u0915\u0930 \u0938\u0942\u091a\u0940 \u092c\u0928\u093e\u090f\u0901\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 9: End-of-Chapter Exercises (\u092a\u094d\u0930\u0936\u094d\u0928 5 - Prime Factorisation):<\/b><\/p>\n            <p><b>\u092a\u094d\u0930\u0936\u094d\u0928 5:<\/b> \u0905\u092d\u093e\u091c\u094d\u092f \u0917\u0941\u0923\u0928\u0916\u0902\u0921\u0928 \u0926\u094d\u0935\u093e\u0930\u093e GCD \u091c\u094d\u091e\u093e\u0924 \u0915\u0930\u0928\u093e\u0964 <br>(i) 180 \u0915\u093e \u0905\u092d\u093e\u091c\u094d\u092f \u0917\u0941\u0923\u0928\u0916\u0902\u0921\u0928 2<sup>2<\/sup> &times; 3<sup>2<\/sup> &times; 5<sup>1<\/sup> \u0939\u0948, \u091c\u093f\u0938\u0947 \u0918\u093e\u0924\u094b\u0902 (Exponents) \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0926\u0930\u094d\u0936\u093e\u092f\u093e \u091c\u093e\u0924\u093e \u0939\u0948\u0964 <br>(ii) \u0926\u094b \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u0940 \u0924\u0941\u0932\u0928\u093e \u0909\u0928\u0915\u0947 \u0905\u092d\u093e\u091c\u094d\u092f \u0906\u0927\u093e\u0930\u094b\u0902 \u0914\u0930 \u0928\u094d\u092f\u0942\u0928\u0924\u092e\/\u0905\u0927\u093f\u0915\u0924\u092e \u0918\u093e\u0924\u094b\u0902 \u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930 \u0915\u0940 \u091c\u093e\u0924\u0940 \u0939\u0948\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 10: \u0905\u092d\u094d\u092f\u093e\u0938 \u0938\u093e\u0930\u093e\u0902\u0936 \u0914\u0930 \u0928\u093f\u0937\u094d\u0915\u0930\u094d\u0937:<\/b><\/p>\n            <p>\u0907\u0938 \u0905\u0927\u094d\u092f\u093e\u092f \u0915\u0947 \u0905\u092d\u094d\u092f\u093e\u0938 \u0939\u092e\u0947\u0902 \u0938\u0930\u0932 \u091c\u094b\u0921\u093c \u0938\u0947 \u0932\u0947\u0915\u0930 \u092f\u0942\u0915\u094d\u0932\u093f\u0921 \u0914\u0930 \u0906\u0930\u094d\u092f\u092d\u091f\u094d\u091f \u0915\u0940 \u0909\u0928\u094d\u0928\u0924 \u0935\u093f\u092d\u093e\u091c\u0928 \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e, \u0905\u092d\u093e\u091c\u094d\u092f \u0917\u0941\u0923\u0928\u0916\u0902\u0921\u0928 \u0914\u0930 \u0921\u0947\u091f\u093e \u0938\u0942\u091a\u093f\u092f\u094b\u0902 \u0915\u0947 \u0935\u094d\u092f\u093e\u0935\u0939\u093e\u0930\u093f\u0915 \u0909\u092a\u092f\u094b\u0917 \u092e\u0947\u0902 \u092a\u093e\u0930\u0902\u0917\u0924 \u092c\u0928\u093e\u0924\u0947 \u0939\u0948\u0902[cite: 13, 14]\u3002<\/p>\n        <\/div>\n    <\/div>\n\n    <div class=\"slider-footer-section\">\n        <button class=\"slider-nav-btn btn-prev\">\u092a\u093f\u091b\u0932\u093e (Prev)<\/button>\n        <span class=\"slider-page-counter\">1 \/ 10<\/span>\n        <button class=\"slider-nav-btn btn-next\">\u0905\u0917\u0932\u093e (Next)<\/button>\n    <\/div>\n<\/div><\/div><\/div><\/div><\/div><div id=\"pg-1239-3\"  class=\"panel-grid panel-no-style\" ><div id=\"pgc-1239-3-0\"  class=\"panel-grid-cell\" ><div id=\"panel-1239-3-0-0\" class=\"widget_text so-panel widget widget_custom_html panel-first-child panel-last-child\" data-index=\"3\" ><div class=\"textwidget custom-html-widget\"><div class=\"pargai-multislider\">\n    <div class=\"slider-header-section\">\n        <div class=\"header-main-title\">\u092d\u093e\u0917 4: \u0905\u0927\u094d\u092f\u093e\u092f 11 - \u0930\u093f\u0935\u093f\u091c\u0928 \u0914\u0930 \u0938\u093e\u0930\u093e\u0902\u0936<\/div>\n    <\/div>\n    \n    <div class=\"slider-content-area\">\n        <div class=\"pargai-slide active-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 1: \u092e\u0941\u0916\u094d\u092f \u092c\u093f\u0902\u0926\u0941 - \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e:<\/b><\/p>\n            <p>\u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u0915\u093f\u0938\u0940 \u0938\u092e\u0938\u094d\u092f\u093e \u0915\u094b \u0939\u0932 \u0915\u0930\u0928\u0947 \u0915\u0940 \u090f\u0915 \u0938\u0941\u0935\u094d\u092f\u0935\u0938\u094d\u0925\u093f\u0924, \u0924\u093e\u0930\u094d\u0915\u093f\u0915 \u0914\u0930 \u091a\u0930\u0923-\u092c\u0926\u094d\u0927 \u092a\u094d\u0930\u0915\u094d\u0930\u093f\u092f\u093e \u0939\u094b\u0924\u0940 \u0939\u0948\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 2: \u092c\u0941\u0928\u093f\u092f\u093e\u0926\u0940 \u091a\u0930\u0923:<\/b><\/p>\n            <p>\u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u0909\u0928 \u092c\u0941\u0928\u093f\u092f\u093e\u0926\u0940 \u091a\u0930\u0923\u094b\u0902 \u0938\u0947 \u092e\u093f\u0932\u0915\u0930 \u092c\u0928\u0924\u0947 \u0939\u0948\u0902 \u091c\u093f\u0928\u094d\u0939\u0947\u0902 \u0939\u092e \u0938\u0940\u0927\u0947 \u0928\u093f\u0937\u094d\u092a\u093e\u0926\u093f\u0924 \u0915\u0930\u0928\u093e \u091c\u093e\u0928\u0924\u0947 \u0939\u0948\u0902\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 3: \u092a\u0941\u0928\u0930\u093e\u0935\u0943\u0924\u094d\u0924\u093f (Iteration):<\/b><\/p>\n            <p>\u0939\u092e \u090f\u0915 \u0939\u0940 \u091a\u0930\u0923 \u0915\u094b \u092c\u093e\u0930-\u092c\u093e\u0930 \u0926\u094b\u0939\u0930\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902, \u091c\u0948\u0938\u0947 1 \u0938\u0947 n \u0924\u0915 \u0915\u0940 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u0947 \u0932\u093f\u090f \u092f\u0939 \u091c\u093e\u0901\u091a \u0915\u0930\u0928\u093e \u0915\u093f \u0915\u094d\u092f\u093e \u0935\u0939 \u0935\u093f\u092d\u093e\u091c\u0915 \u0939\u0948\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 4: \u0938\u0936\u0930\u094d\u0924 \u091a\u0930\u0923 (Conditional Steps):<\/b><\/p>\n            <p>\u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u092e\u0947\u0902 \u0936\u0930\u094d\u0924\u0947\u0902 \u0939\u094b\u0924\u0940 \u0939\u0948\u0902; \u0909\u0926\u093e\u0939\u0930\u0923 \u0915\u0947 \u0932\u093f\u090f, \u0915\u093f\u0938\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u094b \u0935\u093f\u092d\u093e\u091c\u0915 \u0938\u0942\u091a\u0940 \u092e\u0947\u0902 \u0924\u092d\u0940 \u091c\u094b\u0921\u093c\u0947\u0902 \u091c\u092c \u0935\u0939 \u092a\u0942\u0930\u0940 \u0924\u0930\u0939 \u0935\u093f\u092d\u093e\u091c\u093f\u0924 \u0915\u0930\u0924\u0940 \u0939\u094b\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 5: \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u0915\u0940 \u0936\u0941\u0926\u094d\u0927\u0924\u093e:<\/b><\/p>\n            <p>\u091c\u092c \u092d\u0940 \u0939\u092e \u0915\u094b\u0908 \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u092a\u094d\u0930\u0938\u094d\u0924\u0941\u0924 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902, \u0924\u094b \u092f\u0939 \u0924\u0930\u094d\u0915 \u0926\u0947\u0928\u093e \u0906\u0935\u0936\u094d\u092f\u0915 \u0939\u094b\u0924\u093e \u0939\u0948 \u0915\u093f \u0935\u0939 \u0938\u0939\u0940 \u092a\u0930\u093f\u0923\u093e\u092e \u0915\u094d\u092f\u094b\u0902 \u0926\u0947\u0924\u093e \u0939\u0948[cite: 3]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 6: \u0907\u0928\u092a\u0941\u091f \u0914\u0930 \u0926\u0915\u094d\u0937\u0924\u093e:<\/b><\/p>\n            <p>\u0939\u092e \u092f\u0939 \u0935\u093f\u0936\u094d\u0932\u0947\u0937\u0923 \u0915\u0930 \u0938\u0915\u0924\u0947 \u0939\u0948\u0902 \u0915\u093f \u0907\u0928\u092a\u0941\u091f \u0915\u0947 \u0906\u0915\u093e\u0930 \u0915\u0947 \u0906\u0927\u093e\u0930 \u092a\u0930 \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u0915\u093f\u0924\u0928\u0947 \u091a\u0930\u0923 \u0932\u0947\u0924\u093e \u0939\u0948 \u0914\u0930 \u0935\u0939 \u0915\u093f\u0924\u0928\u093e \u0915\u0941\u0936\u0932 \u0939\u0948\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 7: \u0921\u0947\u091f\u093e \u0938\u0902\u0930\u091a\u0928\u093e (Data Structures):<\/b><\/p>\n            <p>\u092c\u0922\u093c\u0924\u0947 \u0915\u094d\u0930\u092e \u092e\u0947\u0902 \u0938\u0942\u091a\u093f\u092f\u093e\u0901 (Lists) \u091c\u0948\u0938\u0947 \u0921\u0947\u091f\u093e \u0938\u094d\u091f\u094d\u0930\u0915\u094d\u091a\u0930 \u091c\u093e\u0928\u0915\u093e\u0930\u0940 \u0915\u094b \u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930 \u0935\u094d\u092f\u0935\u0938\u094d\u0925\u093f\u0924 \u0915\u0930\u0924\u0947 \u0939\u0948\u0902 \u091c\u093f\u0938\u0938\u0947 \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u0905\u0927\u093f\u0915 \u0915\u0941\u0936\u0932 \u092c\u0928\u0924\u093e \u0939\u0948\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 8: \u0906\u0930\u094d\u092f\u092d\u091f\u094d\u091f \u0915\u0940 \u0932\u0902\u092c\u0940 \u092d\u093e\u0917 \u0935\u093f\u0927\u093f:<\/b><\/p>\n            <p>\u092c\u093e\u0930-\u092c\u093e\u0930 \u0918\u091f\u093e\u0928\u0947 \u0915\u0940 \u0924\u0941\u0932\u0928\u093e \u092e\u0947\u0902 '\u0936\u0947\u0937\u092b\u0932 \u0926\u094d\u0935\u093e\u0930\u093e \u0935\u093f\u092d\u093e\u091c\u0928' (Division by remainder) GCD \u0928\u093f\u0915\u093e\u0932\u0928\u0947 \u0915\u0940 \u0905\u0924\u094d\u092f\u0927\u093f\u0915 \u0924\u0940\u0935\u094d\u0930 \u0935\u093f\u0927\u093f \u0939\u0948[cite: 13]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 9: \u0907\u0924\u093f\u0939\u093e\u0938 \u0915\u093e \u092e\u0939\u0924\u094d\u0935:<\/b><\/p>\n            <p>'Algorithm' \u0936\u092c\u094d\u0926 \u092b\u093e\u0930\u0938\u0940 \u0917\u0923\u093f\u0924\u091c\u094d\u091e \u0905\u0932-\u0916\u094d\u0935\u093e\u0930\u093f\u091c\u094d\u092e\u0940 \u0915\u0947 \u0928\u093e\u092e \u0914\u0930 \u092d\u093e\u0930\u0924\u0940\u092f \u0905\u0902\u0915\u0917\u0923\u093f\u0924 \u0915\u0940 \u092a\u0926\u094d\u0927\u0924\u093f\u092f\u094b\u0902 \u0938\u0947 \u0909\u0924\u094d\u092a\u0928\u094d\u0928 \u0939\u0941\u0906 \u0939\u0948[cite: 14]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 10: \u0905\u0927\u094d\u092f\u093e\u092f \u0915\u093e \u0928\u093f\u0937\u094d\u0915\u0930\u094d\u0937:<\/b><\/p>\n            <p>\u092a\u094d\u0930\u093e\u091a\u0940\u0928 \u092d\u093e\u0930\u0924\u0940\u092f \u0917\u0923\u093f\u0924 \u0914\u0930 \u0906\u0927\u0941\u0928\u093f\u0915 \u0915\u0902\u092a\u094d\u092f\u0942\u091f\u0930 \u0935\u093f\u091c\u094d\u091e\u093e\u0928 \u0926\u094b\u0928\u094b\u0902 \u0915\u093e \u0906\u0927\u093e\u0930 \u0938\u0941\u0935\u094d\u092f\u0935\u0938\u094d\u0925\u093f\u0924 \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u0939\u0940 \u0939\u0948\u0902[cite: 14]\u3002<\/p>\n        <\/div>\n    <\/div>\n\n    <div class=\"slider-footer-section\">\n        <button class=\"slider-nav-btn btn-prev\">\u092a\u093f\u091b\u0932\u093e (Prev)<\/button>\n        <span class=\"slider-page-counter\">1 \/ 10<\/span>\n        <button class=\"slider-nav-btn btn-next\">\u0905\u0917\u0932\u093e (Next)<\/button>\n    <\/div>\n<\/div><\/div><\/div><\/div><\/div><div id=\"pg-1239-4\"  class=\"panel-grid panel-no-style\" ><div id=\"pgc-1239-4-0\"  class=\"panel-grid-cell\" ><div id=\"panel-1239-4-0-0\" class=\"widget_text so-panel widget widget_custom_html panel-first-child panel-last-child\" data-index=\"4\" ><div class=\"textwidget custom-html-widget\"><div class=\"pargai-multislider\">\n    <div class=\"slider-header-section\">\n        <div class=\"header-main-title\">\u092d\u093e\u0917 5: \u0905\u0927\u094d\u092f\u093e\u092f 11 - \u0905\u0917\u094d\u0928\u093f\u0935\u0940\u0930 \u0935\u093f\u0936\u0947\u0937 \u0905\u092d\u094d\u092f\u093e\u0938 \u092c\u094d\u0932\u0949\u0915<\/div>\n    <\/div>\n    \n    <div class=\"slider-content-area\">\n        <div class=\"pargai-slide active-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 1: \u0905\u0917\u094d\u0928\u093f\u0935\u0940\u0930 \u0935\u093f\u0936\u0947\u0937 \u092a\u094d\u0930\u0936\u094d\u0928 - \u0936\u092c\u094d\u0926 \u0915\u0940 \u0909\u0924\u094d\u092a\u0924\u094d\u0924\u093f:<\/b><\/p>\n            <p>\u092a\u094d\u0930\u0936\u094d\u0928: 'Algorithm' \u0936\u092c\u094d\u0926 \u0915\u0940 \u0909\u0924\u094d\u092a\u0924\u094d\u0924\u093f \u0915\u093f\u0938 \u092a\u094d\u0930\u0938\u093f\u0926\u094d\u0927 \u0917\u0923\u093f\u0924\u091c\u094d\u091e \u0915\u0947 \u0928\u093e\u092e \u0938\u0947 \u0939\u0941\u0908 \u0939\u0948? \u0909\u0924\u094d\u0924\u0930: \u0905\u0932-\u0916\u094d\u0935\u093e\u0930\u093f\u091c\u094d\u092e\u0940 (Al-Khwarizmi)[cite: 14]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 2: \u0905\u0917\u094d\u0928\u093f\u0935\u0940\u0930 \u0935\u093f\u0936\u0947\u0937 \u092a\u094d\u0930\u0936\u094d\u0928 - \u0910\u0924\u093f\u0939\u093e\u0938\u093f\u0915 \u0917\u094d\u0930\u0902\u0925:<\/b><\/p>\n            <p>\u092a\u094d\u0930\u0936\u094d\u0928: \u0906\u0930\u094d\u092f\u092d\u091f\u094d\u091f \u0915\u0940 \u0915\u093f\u0938 \u092a\u094d\u0930\u0938\u093f\u0926\u094d\u0927 \u0930\u091a\u0928\u093e \u092e\u0947\u0902 \u0932\u0902\u092c\u0940 \u092d\u093e\u0917 \u0935\u093f\u0927\u093f \u0915\u093e \u0909\u0932\u094d\u0932\u0947\u0916 \u0939\u0948? \u0909\u0924\u094d\u0924\u0930: \u0906\u0930\u094d\u092f\u092d\u091f\u0940\u092f (\u0100ryabha\u021b\u012bya, 499 CE)[cite: 13]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 3: \u0905\u0917\u094d\u0928\u093f\u0935\u0940\u0930 \u0935\u093f\u0936\u0947\u0937 \u092a\u094d\u0930\u0936\u094d\u0928 - \u092f\u0942\u0915\u094d\u0932\u093f\u0921 \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u0936\u0930\u094d\u0924:<\/b><\/p>\n            <p>\u092a\u094d\u0930\u0936\u094d\u0928: \u092f\u0942\u0915\u094d\u0932\u093f\u0921 \u092f\u093e \u0906\u0930\u094d\u092f\u092d\u091f\u094d\u091f \u0915\u0947 GCD \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u092e\u0947\u0902 \u092f\u0926\u093f \u0936\u0947\u0937\u092b\u0932 (n) \u0936\u0942\u0928\u094d\u092f \u0939\u094b \u091c\u093e\u090f, \u0924\u094b \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u0915\u094d\u092f\u093e \u0930\u093f\u092a\u094b\u0930\u094d\u091f \u0915\u0930\u0924\u093e \u0939\u0948? \u0909\u0924\u094d\u0924\u0930: \u092a\u0939\u0932\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u092f\u093e \u0905\u0902\u0924\u093f\u092e \u092d\u093e\u091c\u0915 m[cite: 14]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 4: \u0938\u0948\u0928\u094d\u092f \u0905\u0928\u0941\u092a\u094d\u0930\u092f\u094b\u0917 \u0905\u092d\u094d\u092f\u093e\u0938 (HCF):<\/b><\/p>\n            <p>\u092a\u094d\u0930\u0936\u094d\u0928: \u0926\u094b \u091f\u0941\u0915\u0921\u093c\u093f\u092f\u094b\u0902 \u092e\u0947\u0902 \u0938\u0948\u0928\u093f\u0915\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e 825 \u0914\u0930 375 \u0939\u0948\u0964 \u0905\u0927\u093f\u0915\u0924\u092e \u0938\u092e\u093e\u0928 \u0906\u0915\u093e\u0930 \u0915\u0940 \u092a\u0902\u0915\u094d\u0924\u093f\u092f\u093e\u0901 \u092c\u0928\u093e\u0928\u0947 \u092a\u0930 \u092a\u094d\u0930\u0924\u093f \u092a\u0902\u0915\u094d\u0924\u093f \u0938\u0948\u0928\u093f\u0915\u094b\u0902 \u0915\u0940 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u094d\u092f\u093e \u0939\u094b\u0917\u0940? \u0939\u0932: $\\text{HCF}(825, 375) = 75$[cite: 4, 14]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 5: \u0905\u0917\u094d\u0928\u093f\u0935\u0940\u0930 \u0917\u0923\u093f\u0924\u0940\u092f \u0936\u0949\u0930\u094d\u091f\u0915\u091f - LCM \u0914\u0930 HCF:<\/b><\/p>\n            <p>\u092b\u0949\u0930\u094d\u092e\u0942\u0932\u093e: \u0926\u094b \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 A \u0914\u0930 B \u0915\u0947 \u0932\u093f\u090f \u0939\u092e\u0947\u0936\u093e $\\text{HCF} \\times \\text{LCM} = A \\times B$ \u0939\u094b\u0924\u093e \u0939\u0948\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 6: \u0905\u0917\u094d\u0928\u093f\u0935\u0940\u0930 \u0935\u093f\u0936\u0947\u0937 \u092a\u094d\u0930\u0936\u094d\u0928 - \u0905\u092d\u093e\u091c\u094d\u092f \u0938\u0902\u0916\u094d\u092f\u093e \u092a\u0939\u091a\u093e\u0928:<\/b><\/p>\n            <p>\u092a\u094d\u0930\u0936\u094d\u0928: \u0938\u092c\u0938\u0947 \u091b\u094b\u091f\u0940 \u0905\u092d\u093e\u091c\u094d\u092f \u0938\u0902\u0916\u094d\u092f\u093e (Prime Number) \u0915\u094c\u0928 \u0938\u0940 \u0939\u0948? \u0909\u0924\u094d\u0924\u0930: 2 (\u092f\u0939 \u090f\u0915\u092e\u093e\u0924\u094d\u0930 \u0938\u092e \u0905\u092d\u093e\u091c\u094d\u092f \u0938\u0902\u0916\u094d\u092f\u093e \u092d\u0940 \u0939\u0948)\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 7: \u0924\u0930\u094d\u0915 \u0936\u0915\u094d\u0924\u093f \u092a\u094d\u0930\u0936\u094d\u0928 - \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u091a\u0930\u0923:<\/b><\/p>\n            <p>\u092a\u094d\u0930\u0936\u094d\u0928: \u092f\u0926\u093f \u0915\u093f\u0938\u0940 \u091c\u094b\u0921\u093c \u0915\u0940 \u092a\u094d\u0930\u0915\u094d\u0930\u093f\u092f\u093e \u092e\u0947\u0902 \u0905\u0927\u093f\u0915\u0924\u092e \u0905\u0902\u0915 9 \u0914\u0930 9 \u0939\u094b\u0902, \u0924\u094b \u0938\u092c\u0938\u0947 \u0905\u0927\u093f\u0915 \u0938\u0902\u092d\u093e\u0935\u093f\u0924 \u0939\u093e\u0938\u093f\u0932 (Carry) \u0915\u094d\u092f\u093e \u0939\u094b \u0938\u0915\u0924\u0940 \u0939\u0948? \u0909\u0924\u094d\u0924\u0930: 1[cite: 3]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 8: \u0938\u0902\u0916\u094d\u092f\u093e \u092a\u0926\u094d\u0927\u0924\u093f \u0906\u0927\u093e\u0930\u093f\u0924 \u092a\u094d\u0930\u0936\u094d\u0928:<\/b><\/p>\n            <p>\u092a\u094d\u0930\u0936\u094d\u0928: 180 \u0915\u093e \u0905\u092d\u093e\u091c\u094d\u092f \u0917\u0941\u0923\u0928\u0916\u0902\u0921\u0928 \u0930\u0942\u092a \u0915\u094d\u092f\u093e \u0939\u0948? \u0909\u0924\u094d\u0924\u0930: $2^2 \\times 3^2 \\times 5^1$[cite: 14]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 9: \u0907\u0924\u093f\u0939\u093e\u0938 \u0906\u0927\u093e\u0930\u093f\u0924 \u0930\u0915\u094d\u0937\u093e \u092a\u094d\u0930\u0936\u094d\u0928:<\/b><\/p>\n            <p>\u092a\u094d\u0930\u0936\u094d\u0928: 'Liber Algorismi de numero Indorum' \u092a\u0941\u0938\u094d\u0924\u0915 \u0915\u093f\u0938 \u0935\u093f\u0937\u092f \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u0925\u0940? \u0909\u0924\u094d\u0924\u0930: \u092d\u093e\u0930\u0924\u0940\u092f \u0905\u0902\u0915\u0917\u0923\u093f\u0924 \u0914\u0930 \u0938\u094d\u0925\u093e\u0928-\u092e\u093e\u0928 \u092a\u094d\u0930\u0923\u093e\u0932\u0940 \u092a\u0930[cite: 14]\u3002<\/p>\n        <\/div>\n        <div class=\"pargai-slide\">\n            <p><b>\u0938\u094d\u0932\u093e\u0907\u0921 10: \u0905\u0917\u094d\u0928\u093f\u0935\u0940\u0930 \u0938\u093e\u0930\u093e\u0902\u0936 \u0905\u092d\u094d\u092f\u093e\u0938:<\/b><\/p>\n            <p>\u0938\u0948\u0928\u094d\u092f \u090f\u0935\u0902 \u092a\u094d\u0930\u0924\u093f\u092f\u094b\u0917\u0940 \u092a\u0930\u0940\u0915\u094d\u0937\u093e\u0913\u0902 \u092e\u0947\u0902 \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u0914\u0930 \u092e\u0939\u0924\u094d\u0924\u092e \u0938\u092e\u093e\u092a\u0935\u0930\u094d\u0924\u094d\u092f (HCF\/LCM) \u0938\u0947 \u091c\u0941\u0921\u093c\u0947 \u092a\u094d\u0930\u0936\u094d\u0928 \u0924\u093e\u0930\u094d\u0915\u093f\u0915 \u0917\u0924\u093f \u0914\u0930 \u0936\u0941\u0926\u094d\u0927\u0924\u093e \u0915\u094b \u092a\u0930\u0916\u0928\u0947 \u0915\u0947 \u0932\u093f\u090f \u092a\u0942\u091b\u0947 \u091c\u093e\u0924\u0947 \u0939\u0948\u0902[cite: 13]\u3002<\/p>\n        <\/div>\n    <\/div>\n\n    <div class=\"slider-footer-section\">\n        <button class=\"slider-nav-btn btn-prev\">\u092a\u093f\u091b\u0932\u093e (Prev)<\/button>\n        <span class=\"slider-page-counter\">1 \/ 10<\/span>\n        <button class=\"slider-nav-btn btn-next\">\u0905\u0917\u0932\u093e (Next)<\/button>\n    <\/div>\n<\/div><\/div><\/div><\/div><\/div><\/div>","protected":false},"excerpt":{"rendered":"<p>\u092d\u093e\u0917 1: \u0905\u0927\u094d\u092f\u093e\u092f 11 &#8211; \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u0915\u0940 \u0926\u0941\u0928\u093f\u092f\u093e (\u092a\u0930\u093f\u091a\u092f) \u0938\u094d\u0932\u093e\u0907\u0921 1: \u0905\u0927\u094d\u092f\u093e\u092f \u0915\u0940 \u092a\u094d\u0930\u0938\u094d\u0924\u093e\u0935\u0928\u093e: \u0906\u091c\u0915\u0932 &#8216;\u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e&#8217; \u0936\u092c\u094d\u0926 \u0939\u0930 \u091c\u0917\u0939 \u0938\u0941\u0928\u093e\u0908 \u0926\u0947\u0924\u093e \u0939\u0948\u0964 \u0907\u0938 \u0905\u0927\u094d\u092f\u093e\u092f \u092e\u0947\u0902 \u0939\u092e \u0938\u092e\u091d\u0947\u0902\u0917\u0947 \u0915\u093f \u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u0915\u094d\u092f\u093e \u0939\u0948 \u0914\u0930 \u0907\u0938\u0915\u093e \u0917\u0923\u093f\u0924 \u0938\u0947 \u0915\u094d\u092f\u093e \u0938\u0902\u092c\u0902\u0927 \u0939\u0948[cite: 1, 2]\u3002 \u0938\u094d\u0932\u093e\u0907\u0921 2: \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u094b \u0917\u093f\u0928\u0928\u0947 \u0915\u0940 \u092a\u094d\u0930\u093e\u091a\u0940\u0928 \u0935\u093f\u0927\u093f: \u0936\u0941\u0930\u0941\u0906\u0924 \u092e\u0947\u0902 \u0939\u092e \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u094b \u092c\u093f\u0902\u0926\u0941 (dots) [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[10],"tags":[],"class_list":["post-1239","post","type-post","status-publish","format-standard","hentry","category-mathematics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Mathematics &#187; \u0905\u0927\u094d\u092f\u093e\u092f 11 &quot;\u090f\u0932\u094d\u0917\u094b\u0930\u093f\u0926\u092e \u0915\u0940 \u0926\u0941\u0928\u093f\u092f\u093e&quot; &#187; 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