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# Number Theory

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Intern
Joined: 06 Feb 2017
Posts: 2

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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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Joined: 02 Sep 2009
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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?

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