Suppose p and q are integers. Recall that an integer m is even iff m = 2k for some integer k and m is odd iff m = 2k + 1 for some integer k. Prove the following. [You may use the fact that the sum of...

Suppose p and q are integers. Recall that an integer m is even iff m = 2k for some integer k and m is odd iff m = 2k + 1 for some integer k. Prove the following. [You may use the fact that the sum of integers and the product of integers are again integers.]

(a) If p is odd and q is odd, then p + q is even.

(b) If p is odd and q is odd, then pq is odd.

(c) If p is odd and q is odd, then p + 3q is even.

(d) If p is odd and q is even, then p + q is odd.

(e) If p is even and q is even, then p + q is even.

( f ) If p is even or q is even, then pq is even.

(g) If pq is odd, then p is odd and q is odd.

(h) If p²
is even, then p is even.

( i ) If p²
is odd, then p is odd.