3/5 leaves a "remainder" of -2 upon division. (In more complicated "mathspeak" this means .)

Thus is going to leave a remainder of after division, where x is a positive integer. ( .)

And

Now, .

4/5 leaves a remainder of -1 upon division ( ), so leaves a remainder of .

So. Let's finally take a look at this.

divided by 5 will leave a remainder of

Let's do this case by case.

If n is even then

If n is doubly even (ie n = 4x) then .

Now, 16 leaves a remainder of 1 upon division by 5, so

leaves a remainder of upon division by 5. So is divisible by 5 if n is doubly even.

If n is even, but not doubly even then . Then

Dividing this by 5 leaves a remainder of . So is divisible by 5 if n is even but not doubly even.

Thus is divisible by 5 if n is even.

If n is odd then n = 2x + 1:

As in the last case (where n was even but not doubly even) is divisible by 5.

Thus is divisible by 5 if n is odd.

Thus is divisible by 5 for all n.

-Dan