{"id":16055,"date":"2012-12-26T09:01:27","date_gmt":"2012-12-26T16:01:27","guid":{"rendered":"https:\/\/gmatclub.com\/blog\/?p=16055"},"modified":"2012-12-19T16:12:27","modified_gmt":"2012-12-19T23:12:27","slug":"gmat-math-one-is-not-a-prime-number","status":"publish","type":"post","link":"https:\/\/gmatclub.com\/blog\/gmat-math-one-is-not-a-prime-number\/","title":{"rendered":"GMAT Math: One is NOT a Prime Number"},"content":{"rendered":"

\"\"Fact<\/strong>: according to all mathematical definitions, the number 1 is\u00a0not<\/em>\u00a0a prime number<\/p>\n

 <\/p>\n

Primes<\/h2>\n

One definition of a prime number is:\u00a0any number that has only two positive integer factors ---- itself and 1<\/strong>.\u00a0\u00a0 The following are valid prime numbers.<\/p>\n

2, 3, 5, 7, 11, 13, 17, 19, 23, 29 \u2026.<\/p>\n

It is very good to know that 2 is the only even prime number.\u00a0 It is often handy to know the first eight or ten prime numbers.<\/p>\n

Notice that 1 does not fit the fundamental definition for a prime number: it has only one positive integer factor, itself, not two.\u00a0 Therefore, it is not prime.<\/p>\n

 <\/p>\n

Prime factorizations<\/h2>\n

Now, to some folks, that rule will see like a trivial technicality.\u00a0 We permanently excluded Fred from the elite country club because he owns one powder blue tie instead of two.\u00a0 We permanently excluded 1 from the elite set of prime numbers because it has only one positive factor instead of two.\u00a0\u00a0 What kind of arbitrary rip-off is that?\u00a0 Who can we sue?<\/p>\n

It turns out, as usual, mathematicians have deep reasons for the way they draw fine distinctions.\u00a0\u00a0 Let's think about this.\u00a0\u00a0 One major rule of arithmetic is:\u00a0each positive integer greater than one has a unique\u00a0<\/strong>prime factorization<\/strong><\/a>.\u00a0 (This rule is so important, its official name is\u00a0The Fundamental Theorem of Arithmetic<\/a>!)\u00a0\u00a0 The prime factorization of a number is like its DNA: we know its exact constituents, and thus can determine every single one of its factors.\u00a0\u00a0 For example:<\/p>\n

36 = 2*2*3*3<\/p>\n

That is the unique prime factorization of 36, the only way to multiply prime number to get a product of 36.\u00a0\u00a0 Now, pretend for a moment that 1 were a prime number: if we defined things that way, what the consequences would be?\u00a0 Instead of having a unique prime factorization, every number would have\u00a0an infinite number of prime factorizations<\/em>.\u00a0\u00a0 For example, the first few prime factorizations of 36 would be:<\/p>\n

36 = 2*2*3*3<\/p>\n

36 = 1*2*2*3*3\u00a0 (duh<\/em>!)<\/p>\n

36 = 1*1*2*2*3*3\u00a0 (duh<\/em>!)<\/p>\n

36 = 1*1*1*2*2*3*3\u00a0 (duh<\/em>!)\u00a0\u00a0 etc. (an infinite number of\u00a0duh<\/em>! statements)<\/p>\n

Not only would we demolish a perfectly good rule of arithmetic, the Fundamental Theorem of Arithmetic, but in doing so, we also would gain an infinite number of absolutely useless statements.\u00a0\u00a0 That's a lose-lose trade-off!\u00a0\u00a0 Mathematicians, sensing this lose-lose situation, choose it head it off at the pass simply by stating, by fiat, 1 is not a prime number.\u00a0\u00a0 Choosing this particular definition renders irrelevant this troubling situation with an infinite number of\u00a0duh<\/em>! statements as well as numerous similar problematic situations.\u00a0\u00a0 Mathematicians are crafty enough to realize they can avoid a whole boatload of problems just by making a single stipulation: 1 is not a prime number.\u00a0\u00a0 This is the deep reason for the rule.<\/p>\n

 <\/p>\n

Takeaway: 1 is NOT a prime number!<\/strong><\/h2>\n

This post was written by Mike McGarry, GMAT expert at Magoosh<\/a>, and originally posted here<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"

Fact: according to all mathematical definitions, the number 1 is\u00a0not\u00a0a prime number   Primes One definition of a prime number is:\u00a0any number that has only two positive integer factors —-…<\/p>\n","protected":false},"author":133,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9,783,243,736],"tags":[],"class_list":["post-16055","post","type-post","status-publish","format-standard","hentry","category-gmat","category-magoosh-blog","category-blog","category-quant-gmat","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts\/16055","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/users\/133"}],"replies":[{"embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/comments?post=16055"}],"version-history":[{"count":1,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts\/16055\/revisions"}],"predecessor-version":[{"id":16057,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts\/16055\/revisions\/16057"}],"wp:attachment":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/media?parent=16055"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/categories?post=16055"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/tags?post=16055"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}