Q1:

Prove or disprove: for every positive integer n, the congruence equation

must hold.

Here are my thoughts:

A1: Let n be any positive integer. Suppose

does not hold. Thus, it is not the case that

. Now, notice that

. We can see that

is even whenever

is odd and vice-versa. Hence, either expression will at least be divisible by two, which is a factor of 6. Thus,

which is the same as

. This contradicts our original claim.

I don't know much about modular arithmetic, so I am not sure if I am doing this right.