Answer to Question #313 in Algebra for Georgia

Question

Prove that the square of any prime number p > 3 when divided by 12 gives modulo 1.

Answer 1

Consider any possible modulo when dividing a prime number p > 3 by 6. It can not give modulo 2 or 4, because then it would be even. It may not give modulo 3, because then it would be divisible by 3. Thus, a prime number p > 3 when dividing by 6 gives modulo 1 or 5, ie it looks like 6n ± 1; and its square looks like 36n² ± 12n + 1.