IB HL Math

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I need to that if there is better way to prove the following:
I am trying to prove that
x
r
= k
and k is a multiple of x only when x is prime.

I said that if x is non-prime, then:
Let a = 6
Let r = 4

6!
(6 – 4)! 4!

= 3(2) x 5 x 4!
(2!) 4!
= 15
It is not divisible by 6

My explanation is:
When a is a non-prime number, then a! is [a x (a-1) x (a-2)…x 2 x 1] and a divides out with at least one of the r and this makes it possible for k to be not divisible by x.


and when x is prime:
For example,
Let a = 7
Let r = 4
7!
(7 – 4)! 4!

= 7 x 6 x 5 x 4!
(3!) 4!

= 35
It is divisible by 7

My explanation is:
When a is a prime number, then a! is [a x (a-1) x (a-2)…x 2 x 1] and a does not divide out with any r and this makes it possible for k to be divisible by x.


I am sure there is an elegant wa to prove this; however, i don't exactly know how to do this...can u help plz?? i really appreciate it..thnx

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