{"id":89,"date":"2022-12-15T10:00:38","date_gmt":"2022-12-15T04:30:38","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=89"},"modified":"2022-12-16T09:26:24","modified_gmt":"2022-12-16T03:56:24","slug":"fundamental-theorem-of-arithmetic","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/fundamental-theorem-of-arithmetic\/","title":{"rendered":"What Is Fundamental Theorem of Arithmetic"},"content":{"rendered":"

Fundamental Theorem of Arithmetic<\/strong><\/h2>\n

We have discussed about Euclid Division Algorithm<\/a> in the previous post.<\/p>\n

Fundamental 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>\n

In 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>\n

Fundamental Theorem of Arithmetic\u00a0<\/strong>Example Problems With Solutions<\/strong><\/h2>\n

Example 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>\n

Example 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":"\nWhat Is Fundamental Theorem of Arithmetic - A Plus Topper<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.aplustopper.com\/fundamental-theorem-of-arithmetic\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"What Is Fundamental Theorem of Arithmetic\" \/>\n<meta property=\"og:description\" content=\"Fundamental Theorem of Arithmetic We have discussed about Euclid Division Algorithm in the previous post. 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