These definitions help in solving this problem
Even Number:
A number m is even iff there exists an integer k such that m = 2k
Odd Number:
A number m is odd iff there exists an integer k such that m = 2k+1
(a) If n and m are both odd, then n + m is even.
n = 2p+1 and m = 2q+1 (p,q are integers)
n+m = 2p+1+2q+1 = 2(p+q+1) = 2(some integer) = even no
(b) If n is odd and m is even, then n + m is odd.
n = 2p+1 and m = 2q (p,q are integers)
n+m = 2p+1+2q = 2(p+q)+1 = 2(some integer)+1 = odd no
(c) If n and m are both even, then n + m is even
n = 2p and m = 2q (p,q are integers)
n+m = 2p+2q = 2(p+q) = 2(some integer) = even no
(d) If n and m are both odd, then nm is odd; otherwise, nm is even.
Both n & m are odd
n = 2p+1 and m = 2q+1 (p,q are integers)
n.m = (2p+1).(2q+1) = 4pq+2p+2q+1 = 2(2pq+p+q)+1 = 2(some integer)+1 = odd no
Both n & m are even
n = 2p and m = 2q (p,q are integers)
n.m = 4pq = even number