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Number Theory

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Intern
Joined: 06 Feb 2017
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14 Sep 2017, 21:49
19! - 11 is a prime number? And is there any properties relating factorials and prime numbers?

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15 Sep 2017, 08:02
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Ayush12 wrote:
19! - 11 is a prime number? And is there any properties relating factorials and prime numbers?

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A Prime number is a positive integer with exactly two distinct divisors: 1 and itself.

Now, we can factor out 11 from 19! - 11 to get $$11(1*2*3*4*5*6*7*8*9*10*12*13*14*15*16*17*18*19- 1)$$, so we know that 19! - 11 does not have only 1 and 19! - 11 as its factors (it has at lest two more factors 11 and 1*2*3*4*5*6*7*8*9*10*12*13*14*15*16*17*18*19- 1), so 19! - 11 is NOT a prime.
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19 Sep 2017, 17:17
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Ayush12 wrote:
19! - 11 is a prime number? And is there any properties relating factorials and prime numbers?

Sent from my MotoG3 using GMAT Club Forum mobile app

You're probably thinking of problems that ask you to factor a number like 20! + 12, or 11! + 7, etc. In general, there are a few rules that apply here:

- to factor a sum of two numbers, start by looking for factors they have in common. If a and b share a factor, then that number also factors into a+b.
- a factorial, like 20!, is a product of a whole list of smaller numbers: 20! = 20*19*18*17*...*3*2*1. That means every number from 1 on up to 20 is definitely a factor of 20!. There are also other, larger factors, but those ones are the easiest ones to find.

So if you want to factor 20!+12, for example, look for factors that 20! and 12 have in common. 12 is a factor of both 20! and 12. So, 12 is a factor of 20!+12.

For a somewhat tougher problem - how about 20! + 27? It's not immediately obvious whether 27 is a factor of 20!. However, look at how 20! breaks down one more time: 20! = 20 * 19 * 17 * 18 * ... * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1. Since 9*3 divides into 20!, 27 is a factor of 20!. So, 27 is a factor of 20! + 27.

You can use a similar approach on other problems, such as the one you posed here.
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Chelsey Cooley | Manhattan Prep Instructor | Seattle and Online

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Re: Number Theory   [#permalink] 19 Sep 2017, 17:17
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