Question

If a number is divisible 6 then it must be divisible by 12

Answer 1

For instance, 36 is divisible by both 6 and 12; however, 42 is divisible by 6 but not by 12.

Let N be a a positive integer which is divisible by 6. Then N = 6*k, where k is a positive integer. So N = (12/2)*k = 12*(k/2). So N is a multiple of 12 if and only is k/2 is an integer. So N is a multiple of 12 if and only if k is a multiple of 2, that is if and only if k is even.

So:

A positive integer N is divisible by 6 if and only if N/6 is a positive integer.

A positive integer N is divisible by 12 if and only if N/6 is an even positive integer.

From this, and the fact that the set of even positive integers are a proper subset of the set of positive integers, it follows that the set of positive integers which are divisible by 12 form a proper subset of the set of positive integers which are divisible by 6. So:

Every positive integer which is divisible by 12 is divisible by 6, but there are positive integers (e.g. 6, 18, 66) which are divisible by 6 but are not divisible by 12.