- #1

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## Homework Statement

For any integers a and b and any positive integers k and j, if ##a \equiv 2-b \pmod{k}## and ##j \mid k##, then ##a^2 + 4b - b^2 \equiv 4 \pmod{j}##

## Homework Equations

##x \equiv y\pmod{q}## then q|x-y

## The Attempt at a Solution

At first I thought this would probably be straight forward and easy. But I got stuck a couple steps in:

if ##a \equiv 2-b \pmod{k}## then ##k \mid a-2-b## and ##a-b-2=kn_o## (where ##n_o## is some integer). Also, if ##j \mid k## then ##k=jn_i## (where ##n_i## is an integer).

(magic)

I know that finally my answer should look like ##a^2 + 4b - b^2 \equiv 4 \pmod{j}## which is equivalent to ##a^2+4b-b^2-4=jn_x##

I'm having trouble with the magic part. I know since ##k=jn_i## I can rewrite ##a-b-2=kn_o## as ##a-b-2=jn_in_o## and then replace ##n_o n_i## with ##n##. But I don't see what I should do next. It seems to me that the ##a-b-2## should get squared but I end up with too many a's when I do so. Anyway, I don't yet see the reasoning behind squaring the value to begin with.

If anyone has a suggestion for this I'd be very appreciative.