Proof using primes, divisibility, and sum of squares

[ccsmarty]

Homework Statement

I have to prove or disprove the following:

Part a) If p is prime and p | (a² + b²) and p | (c² + d²), then p | (a² - c²)

Part b) f p is prime and p | (a² + b²) and p | (c² + d²), then p | (a² + c²)

Homework Equations

The Attempt at a Solution

Part a)

Since p | (a² + b²), we have that a² + b² = pk, for some integer k.
Since p | (c² + d²), we have that c² + d² = pt, for some integer t.

Suppose p | (a² - c²), then we have that a² - c² = pr, for some integer r.

By solving for a² and c² in the above equations, and substitution we have that
pk - b² - (pt - d²) = pr
pk - pt - b² + d² = pr
pk - pt - pr = b² - d²
p (k - t - r) = b² - d²
So p | (b² - d²)

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...

 

[Dick]

Did you try looking for counterexamples before going CRAZY? That's always a good idea.