Here's a hint to get you started. An integer that is congruent to 3(mod 4) can be written in the form 4k+3 where k is an integer.
I can show the following proofs for specific examples, but can't do the actual proof itself. Any way of starting would be great. I thought about induction, but I can't understand how that would work.
Show that the product of 2 primes that are both congruent 3 (mod 4) is congruent 1 (mod 4)
Show that the product of any no. of primes that are congruent 1 (mod 4) is congruent 1 (mod 4)
Show that the product of any no. of odd primes is congruent 1 (mod 4), unless an odd no. of the primes are congruent 3 (mod 4)
thanks
I think you can do the last one by mathematical induction. You will need to prove that when you multiply an odd integer congruent to 1 (mod 4) by an odd integer congruent to 3 (mod 4) you get an odd integer congruent to 3 (mod 4) (this is easy if you understood the first 2). Use this result together with your first two questions.