Euclid Division Algorithm<\/a> in the previous post.<\/p>\nFundamental Theorem of Arithmetic:<\/strong> \nStatement:<\/strong> Every composite number can be decomposed as a product prime numbers in a unique way, except for the order in which the prime numbers occur. \nFor example: \n(i) \u00a030 = 2 \u00d7 3 \u00d7 5, 30 = 3 \u00d7 2 \u00d7 5, 30 = 2 \u00d7 5 \u00d7 3 and so on. \n(ii) 432 = 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 3 \u00d7 3 \u00d7 3 = 24<\/sup> \u00d7 33<\/sup> \nor 432 = 33<\/sup> \u00d7 24<\/sup>. \n(iii) 12600 = 2 \u00d7 2 \u00d7 2 \u00d7 3 \u00d7 3 \u00d7 5 \u00d7 5 \u00d7 7 \n= 23<\/sup> \u00d7 32<\/sup> \u00d7 52<\/sup> \u00d7 7<\/p>\nIn general, a composite number is expressed as the product of its prime factors written in ascending order of their values. \nExample: (i) 6615 = 3 \u00d7 3 \u00d7 3 \u00d7 5 \u00d7 7 \u00d7 7 \n= 33<\/sup> \u00d7 5 \u00d7 72<\/sup> \n(ii) 532400 = 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 5 \u00d7 5 \u00d7 11 \u00d7 11 \u00d7 11<\/p>\nFundamental Theorem of Arithmetic\u00a0<\/strong>Example Problems With Solutions<\/strong><\/h2>\nExample 1: \u00a0 \u00a0<\/strong>Consider the number 6n<\/sup>, where n is a natural number. Check whether there is any value of n \u2208 N for which 6n <\/sup>is divisible by 7. \nSol. \u00a0 \u00a0<\/strong>Since, \u00a0 6 = 2 \u00d7 3; 6n<\/sup> = 2n<\/sup> \u00d7 3n<\/sup> \n\u21d2 The prime factorisation of given number 6n<\/sup> \n\u21d2 6n<\/sup> is not divisible by 7.<\/strong><\/p>\nExample 2:<\/strong> \u00a0 Consider the number 12n<\/sup>, where n is a natural number. Check whether there is any value of n \u2208<\/strong>\u00a0N for which 12n<\/sup> ends with the digit zero. \nSol.<\/strong>\u00a0 \u00a0 \u00a0We know, if any number ends with the digit zero it is always divisible by 5. \nIf 12n<\/sup> ends with the digit zero, it must be divisible by 5. \nThis is possible only if prime factorisation of 12n<\/sup> contains the prime number 5. \nNow, 12 = 2 \u00d7 2 \u00d7 3 = 22<\/sup> \u00d7 3 \n\u21d2 12n<\/sup> = (22<\/sup> \u00d7 3)n<\/sup> = 22n<\/sup> \u00d7 3n<\/sup> \ni.e., prime factorisation of 12n<\/sup> does not contain the prime number 5. \n\u21d2 There is no value of n \u2208<\/strong>\u00a0N for which\u00a0<\/strong>12n<\/sup> ends with the digit zero.<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"Fundamental Theorem of Arithmetic We have discussed about Euclid Division Algorithm in the previous post. Fundamental Theorem of Arithmetic: Statement: Every composite number can be decomposed as a product prime numbers in a unique way, except for the order in which the prime numbers occur. For example: (i) \u00a030 = 2 \u00d7 3 \u00d7 5, […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[5],"tags":[14,15,7],"yoast_head":"\n
What Is Fundamental Theorem of Arithmetic - A Plus Topper<\/title>\n \n \n \n \n \n \n \n \n \n \n \n \n \n\t \n\t \n\t \n