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Math Help - prime and modulo proofs

  1. #1
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    prime and modulo proofs

    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
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  2. #2
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    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.
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  3. #3
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    I got the first 2 done by the hint. But I still can't seem to get my head around the third one.
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  4. #4
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    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.
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