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 is even, then we can write it in a form two times another integer, say . That is,
Now, looking at condition 2, a number is divisible by 6 if it is also divisible by 3. This means that must be divisible by since is divisible by 3. Now, if is divisible by , then we can write it as 3 multiplied by another integers, say, . That is,
Combining the first and second equations, we have
This means that which is clearly divisible by 6.