{"id":3167,"date":"2012-12-11T10:46:10","date_gmt":"2012-12-11T18:46:10","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=3167"},"modified":"2020-01-15T10:50:14","modified_gmt":"2020-01-15T18:50:14","slug":"gmat-math-one-is-not-a-prime-number","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/gmat-math-one-is-not-a-prime-number\/","title":{"rendered":"GMAT Math: Is 1 a Prime Number?"},"content":{"rendered":"

\"\"<\/p>\n

Spoiler alert<\/strong>: according to all mathematical definitions, the number 1 is not<\/em> a prime number<\/p>\n

Primes<\/h2>\n

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

2, 3, 5, 7, 11, 13, 17, 19, 23, 29 \u2026 \u221e<\/center><\/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

Prime Factorizations<\/h2>\n

Now, to some folks, that rule will seem like a trivial technicality.\u00a0 <\/p>\n

We permanently excluded Fred from the elite country club because he owns one powder blue tie instead of two.\u00a0 <\/p>\n

We permanently excluded 1 from the elite set of prime numbers because it has only one positive factor instead of two.\u00a0\u00a0 <\/p>\n

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: each positive integer greater than one has a unique <\/strong>prime factorization<\/strong><\/a>.\u00a0 (This rule is so important, its official name is The Fundamental Theorem of Arithmetic<\/a>!)\u00a0\u00a0 <\/p>\n

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.<\/p>\n

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 numbers to get a product of 36.\u00a0\u00a0 <\/p>\n

Now, pretend for a moment that 1 were a prime number: if we defined things that way, what would the consequences be?\u00a0 <\/p>\n

Instead of having a unique prime factorization, every number would have an infinite number of prime factorizations<\/em>.\u00a0\u00a0 <\/p>\n

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 duh<\/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 <\/p>\n

That’s a lose-lose trade-off!\u00a0\u00a0 Mathematicians, sensing this lose-lose situation, choose to head it off at the pass simply by stating, by fiat, 1 is not a prime number.\u00a0\u00a0 <\/p>\n

Choosing this particular definition renders irrelevant this troubling situation with an infinite number of duh<\/em>! statements as well as numerous similar problematic situations.\u00a0\u00a0 <\/p>\n

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

Takeaway: 1 is NOT a prime number!<\/strong><\/h2>\n","protected":false},"excerpt":{"rendered":"

Spoiler alert: according to all mathematical definitions, the number 1 is not a prime number Primes One definition of a prime number is: any number that has only two positive integer factors—itself and 1.\u00a0\u00a0 The following are valid prime numbers. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 \u2026 \u221e It is very […]<\/p>\n","protected":false},"author":26,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[160],"tags":[],"ppma_author":[13209],"class_list":["post-3167","post","type-post","status-publish","format-standard","hentry","category-arithmetic"],"acf":[],"yoast_head":"\nGMAT Math: Is 1 a Prime Number? - Magoosh Blog \u2014 GMAT\u00ae Exam<\/title>\n<meta name=\"description\" content=\"Is 1 a prime number? Nope, 1 is neither prime nor composite. 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He was also featured as \"member of the month\" for over two years at <a href=\"https:\/\/gmatclub.com\/blog\/2012\/09\/mike-mcgarrys-gmat-experience\/\" rel=\"noopener noreferrer\">GMAT Club<\/a>. Mike holds an A.B. in Physics (graduating <em>magna cum laude<\/em>) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's <a href=\"https:\/\/www.youtube.com\/c\/MagooshGMATChannel\/featured\" rel=\"noopener noreferrer\">Youtube <\/a>video explanations and resources like <a href=\"https:\/\/magoosh.com\/gmat\/whats-a-good-gmat-score\/\" rel=\"noopener noreferrer\">What is a Good GMAT Score?<\/a> and the <a href=\"https:\/\/magoosh.com\/gmat\/gmat-diagnostic-test\/\" rel=\"noopener noreferrer\">GMAT Diagnostic Test<\/a>."}],"_links":{"self":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/3167","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/users\/26"}],"replies":[{"embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/comments?post=3167"}],"version-history":[{"count":0,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/3167\/revisions"}],"wp:attachment":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/media?parent=3167"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/categories?post=3167"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/tags?post=3167"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/ppma_author?post=3167"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}