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Use any proof strategy to solve the following sub-problems. (a) Prove that for integers a and b, if a+b2 19, then a 2 10 orb 2 10. (b) Prove that there are no positive integer solutions to the equation x2-2 (c) In mathematics, "X y" denotes x divides y, namely x is a factor of y, or in other words, y s divisible by x. So if sly, we can write y as. k, where k is also a factor of y. Prove that for integers a and b, if a > 2,then at b or at (b+1). (d) Prove that for integers a and b,if 5tab, then 5ta and 51b. (e) Consider the following definition: An integer n is sane is 3(n +2n) Give a counterexample to the following: All odd integers are sane. Give a direct proof of the following: If 3 n, then n is sane. Prove by contradiction: If n 3j+2 for some integer j, then n is not sane. ttu