Let be the statement that is divisible by .

is the staement that is divisible by .

is true.

Assume is true, where is positive integer .

Prove that must be true.

for some integer as is true.

Observe the expression, it is clearly divisible by as and are integers.

The final statement is exactly , so is true.

Because is true for and implies , is true for and so on.

By the principle of mathematical induction, is divisible by for all positive integers .

is divisible by for all positive integers .