Prove that if m and n are even integers, then mn+ 1-m-n is an odd integer.

5) Prove that the sum of two fourth powers of integers can never be three more than a multiple of 4. In other words, prove that it is not 4 4 possible for n +m =4 k + 3, for integers n, m and k. [Hint: Consider fourth powers of even and odd integers.]

Show that for every integer n ≥ 8, there exist integers a ≥ 0 and b ≥ 0 such that n = 3a + 5b.

prove n2 + 5 greater than or equal to 4n for all integers n