Prove: 1. There exists an integer n, so that n³ - n is odd. 2. For all a, b E Z†, if a > 1 and b > 1, then gcd (a, b) < gcd (a³,b³) (if false, prove by negation)

(a) Assume that n1, n2 and n3 are integers with ged(n1, n2) = gcd(n2, n3) = gcd(n1, n3) = 1. Let aj, a2 and az be integers and let N1 = n2n3, N2 = n,n3 and N3 = n¡n2. Consider the integer z = a,N1) + azNw2) + azN(ns). Prove that r = a1 mod n1, I = a2 mod ng and r = a3 mod n3 (b) Use part (a) to find a solution to the system of congruences I = 3 mod 8, I = 2 mod 3, r = 1 mod 5

If p,q,r are prime number but p is not equal to q, then what is gcd(p,q)? What is the smallest positive integer of the form px + py where x, y ∈ Z?