PLease explain and show work if possible thank you. Let p be a prime number and

Question

PLease explain and show work if possible thank you.

Let p be a prime number and a, b, c positive integers. For each of the following statements, state if it is true or false. If it is true, give a proof. If it is false, give at least one counterexample. If p|a and p|(a2 + b2), then p. If p|an, n 1, then p|a. If p|(a2 + b2) and p|(b2 + c2) then p|(a2 - c2) If p|(a2 + b2) and p|(b2 + c2), then p|(a2 + c2).

Explanation / Answer

a)true, p|a => p|a^2, as p|a^2+b^2 also p|(a^2+b^2 - a^2) => p|b^2 b)true, suppose p does not divide a, then p|a^2, then p|a^3 ... thus p|a^n a contradiction. So statement b) is true. c)true, p|a^2 + b^2 and p|b^2 + c^2 => p| a^2 + b^2 - b^2 + c^2 = > p|a^2 - c^2 d) false, 5 is a prime. 5| 8^2 + 1^2, 5 | 1^2 + 2^2 but 5 does not divide 8^2 + 2^2 = 64 + 4 = 68. thus we are done.

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