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## Homework Statement

I have to prove or disprove the following:

Part a) If p is prime and p | (a

^{2}+ b

^{2}) and p | (c

^{2}+ d

^{2}), then p | (a

^{2}- c

^{2})

Part b) f p is prime and p | (a

^{2}+ b

^{2}) and p | (c

^{2}+ d

^{2}), then p | (a

^{2}+ c

^{2})

## Homework Equations

## The Attempt at a Solution

Part a)

Since p | (a

^{2}+ b

^{2}), we have that a

^{2}+ b

^{2}= pk, for some integer k.

Since p | (c

^{2}+ d

^{2}), we have that c

^{2}+ d

^{2}= pt, for some integer t.

Suppose p | (a

^{2}- c

^{2}), then we have that a

^{2}- c

^{2}= pr, for some integer r.

By solving for a

^{2}and c

^{2}in the above equations, and substitution we have that

pk - b

^{2}- (pt - d

^{2}) = pr

pk - pt - b

^{2}+ d

^{2}= pr

pk - pt - pr = b

^{2}- d

^{2}

p (k - t - r) = b

^{2}- d

^{2}

So p | (b

^{2}- d

^{2})

I don't know where to go from here.

I figure that once I figure out how to do this part, the second part should be very similar.

Any help would be greatly appreciated. I'm going CRAZY trying to figure this out...