a) Prove that the sum of any two odd integers is an even integer.b) Prove that the sum of an odd integer and an even integer is always odd.c) Let m and n be integers such that m is even. Prove that the product mn is even.

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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