{"id":36422,"date":"2025-06-25T07:17:02","date_gmt":"2025-06-25T07:17:02","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=36422"},"modified":"2025-06-25T07:17:03","modified_gmt":"2025-06-25T07:17:03","slug":"find-the-prime-factorisation-of-each-of-the-following-numbers-leaving-your-answer-in-index-notation","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/find-the-prime-factorisation-of-each-of-the-following-numbers-leaving-your-answer-in-index-notation\/","title":{"rendered":"Find the prime factorisation of each of the following numbers, leaving your answer in index notation."},"content":{"rendered":"\n<p>Find the prime factorisation of each of the following numbers, leaving your answer in index notation. (a) 8624 (b) 6804 (c) 26 163 (d) 196 000<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To find the <strong>prime factorisation<\/strong> of a number, we break it down into <strong>prime numbers<\/strong> (like 2, 3, 5, 7, 11, etc.) that multiply together to give the original number. When expressing the result in <strong>index notation<\/strong>, we use <strong>exponents<\/strong> to show how many times a prime number repeats in the factorisation.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>(a) 8624<\/strong><\/h3>\n\n\n\n<p>Start dividing by the smallest prime number:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>8624 \u00f7 2 = 4312<\/li>\n\n\n\n<li>4312 \u00f7 2 = 2156<\/li>\n\n\n\n<li>2156 \u00f7 2 = 1078<\/li>\n\n\n\n<li>1078 \u00f7 2 = 539<\/li>\n\n\n\n<li>539 \u00f7 7 = 77<\/li>\n\n\n\n<li>77 \u00f7 7 = 11<\/li>\n\n\n\n<li>11 \u00f7 11 = 1<\/li>\n<\/ul>\n\n\n\n<p>So, the prime factors:<br><strong>2\u2074 \u00d7 7\u00b2 \u00d7 11<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>(b) 6804<\/strong><\/h3>\n\n\n\n<p>Start with 2:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>6804 \u00f7 2 = 3402<\/li>\n\n\n\n<li>3402 \u00f7 2 = 1701<br>Now try 3:<\/li>\n\n\n\n<li>1701 \u00f7 3 = 567<\/li>\n\n\n\n<li>567 \u00f7 3 = 189<\/li>\n\n\n\n<li>189 \u00f7 3 = 63<\/li>\n\n\n\n<li>63 \u00f7 3 = 21<\/li>\n\n\n\n<li>21 \u00f7 3 = 7<\/li>\n<\/ul>\n\n\n\n<p>So, the prime factors:<br><strong>2\u00b2 \u00d7 3\u2075 \u00d7 7<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>(c) 26163<\/strong><\/h3>\n\n\n\n<p>Check for divisibility by 3:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>2 + 6 + 1 + 6 + 3 = 18 \u2192 divisible by 3<\/li>\n\n\n\n<li>26163 \u00f7 3 = 8721<\/li>\n\n\n\n<li>8721 \u00f7 3 = 2907<\/li>\n\n\n\n<li>2907 \u00f7 3 = 969<\/li>\n\n\n\n<li>969 \u00f7 3 = 323<br>Now try 17:<\/li>\n\n\n\n<li>323 \u00f7 17 = 19<\/li>\n<\/ul>\n\n\n\n<p>So, the prime factors:<br><strong>3\u2074 \u00d7 17 \u00d7 19<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>(d) 196000<\/strong><\/h3>\n\n\n\n<p>Start with 10:<br>196000 = 196 \u00d7 1000<\/p>\n\n\n\n<p>Break each part:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>196 = 2\u00b2 \u00d7 7\u00b2<\/li>\n\n\n\n<li>1000 = 10\u00b3 = (2 \u00d7 5)\u00b3 = 2\u00b3 \u00d7 5\u00b3<\/li>\n<\/ul>\n\n\n\n<p>Now multiply all together:<br><strong>2\u00b2 \u00d7 7\u00b2 \u00d7 2\u00b3 \u00d7 5\u00b3<\/strong><br>Combine like bases:<br><strong>2\u2075 \u00d7 5\u00b3 \u00d7 7\u00b2<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation <\/h3>\n\n\n\n<p>Prime factorisation is the process of writing a number as a product of prime numbers. Prime numbers are those greater than 1 with only two factors \u2014 1 and themselves. Every composite number has a unique prime factorisation, which makes this process valuable in arithmetic, algebra, and number theory.<\/p>\n\n\n\n<p>The factorisation is done by dividing the number repeatedly by the smallest prime number possible (starting with 2, then 3, 5, etc.) until we reach 1. Whenever a prime number divides evenly into the number, we record it and divide the result again. If it does not divide evenly, we move to the next higher prime number.<\/p>\n\n\n\n<p>For large numbers, breaking them into smaller known values (like 1000 = 10\u00b3) can simplify the work. Once we finish breaking down the number into all its prime components, we collect like bases and express the count of each as an exponent. This is called index notation. For example, instead of writing 2 \u00d7 2 \u00d7 2 \u00d7 2 = 16, we write 2\u2074.<\/p>\n\n\n\n<p>Using index notation helps simplify expressions, especially when working with large or repeated prime factors. It also makes it easier to compare factorizations or compute greatest common divisors and least common multiples.<\/p>\n\n\n\n<p>Understanding prime factorisation is also essential in cryptography, data encoding, and solving mathematical problems involving divisibility, fractions, and ratios.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"852\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-792.jpeg\" alt=\"\" class=\"wp-image-36429\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-792.jpeg 852w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-792-250x300.jpeg 250w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-792-768x923.jpeg 768w\" sizes=\"auto, (max-width: 852px) 100vw, 852px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Find the prime factorisation of each of the following numbers, leaving your answer in index notation. (a) 8624 (b) 6804 (c) 26 163 (d) 196 000 The Correct Answer and Explanation is: To find the prime factorisation of a number, we break it down into prime numbers (like 2, 3, 5, 7, 11, etc.) that [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-36422","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/36422","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=36422"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/36422\/revisions"}],"predecessor-version":[{"id":36430,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/36422\/revisions\/36430"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=36422"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=36422"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=36422"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}