# IB HL Math

posted by
**Asad**
.

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