Proof of Divisibility By 6 - Proofs from The Book

After discussing divisibility by 5, we proceed to divisibility by 6. A number is divisible by 6 if

(1) it is even
(2) it is divisible by 3

The explanation to this is quite simple. First, if a number is even, then it is divisible by 2.

Let n be that number. Since $n$ is even, then we can write it in a form two times another integer, say $k$ . That is,

$n = 2k$ .

Now, looking at condition 2, a number is divisible by 6 if it is also divisible by 3. This means that $k$ must be divisible by $3$ since $2k$ is divisible by 3. Now, if $k$ is divisible by $3$ , then we can write it as 3 multiplied by another integers, say, $h$ . That is,