EXPERT ANSWER
definition of even integer: any integer m is called even if m = 2k for some integer k
definition of odd integer: any integer n is called odd if n = 2l+1 for some integer l
(a)Suppose n and m are odd integers.
Then n = 2k + 1 and m = 2l + 1 for some k; l belonging to Z, by the denition of an odd integer.
Therefore n + m = (2k + 1) + (2l + 1) = 2(k + l + 1).
Since k and l are integers, so is k + l + 1.
Hence n + m = 2p with p = k + l + 1 2 Z.
By the denition of an even integer, this shows that n + m is even.
(b)Suppose n is an odd integer and m is an even integer.
Then n = 2k + 1 and m = 2l for some k; l belonging to Z, by the denition of an odd integer and even integer.
Therefore n + m = (2k + 1) + (2l) = 2(k + l)+ 1.
Since k and l are integers, so is k + l .
Hence n + m = 2p + 1 with p = k + l belonging to Z.
By the denition of an odd integer, this shows that n + m is odd.
(c)suppose m is even integer and n is any other integer
then m = 2k for some k belonging to Z, by the definition of even integer
mn = (2k)(n) = 2(kn)
k and n are integers so is the product kn
therefore from the definition of even integer this shows that mn is even