If x is congruent to a modulo n then either x is congruent to a modulo 2n or...

Prove the following conditional; If then either or

So just fiddling with x's and a's...

These all look like good conditionals to me so I'm satisfied that I can write a proof.

I've tried a direct proof where I just multiply by 2 but that doesn't seem too profitable.

My other idea is to assume that but that doesn't seem to work out well either.

And

Neither of which seem too promising.

Any ideas or errors?

Re: If x is congruent to a modulo n then either x is congruent to a modulo 2n or...

Hi,

Hint: From $x\equiv a\pmod n$, there is $q\in\mathbb Z$ with $x-a=nq$. Consider the two cases q even and q odd.

Re: If x is congruent to a modulo n then either x is congruent to a modulo 2n or...

Assume

Since q is either even or odd,

OK that looks great thanks!