Prove that an integer n is divisible by 6 if and only if it

Prove that an integer n is divisible by 6 if and only if it is divisible by both 2 and 3.

I am looking for a very clear and well explained proof. I\'m more interested in understanding the proof than the end result.

Solution

Let n be an integer divisible by 6.

By definition n = 6k for some integer k.

Thus n = 2 *3*k = 3*(2k).

But 2k is an integer since it is the product of integers.

Thus n = 3*(an integer). Hence by definition of divides 3

hence an integer n is divisible by 6 if and only if it is divisible by both 2 and 3.