1. a Show that if p mod 4 = 3, there is no integer n such that p divides n2 + 1.
b But show that if p mod 4 = 1, there is such an integer.
2. Consider two fractions m/n and m’/n’ in lowest terms. Prove that when the sum m/n+m’/n’ is reduced to lowest terms, the denominator will be nn’ if and only if n ⊥ n’ . (In other words, (mn’+m’n)/nn’ will already be in lowest terms if and only if n and n’ have no common factor.)