Prove that
(a) there exist integers m and n such that 2m + 7n = 1.
(b) there exist integers m and n such that 15m + 12n = 3.
(c) there do not exist integers m and n such that 2m + 4n = 7.
(d) there do not exist integers m and n such that 12m + 15n = 1.
(e) for every integer t, if there exist integers m and n such that 15m+16n=t, then there exist integers r and s such that 3r + 8s = t.
(f) if there exist integers m and n such that 12m + 15n = 1, then m and n are both positive.
(g) for every odd integer m, if m has the form 4k + 1 for some integer k, then m + 2 has the form 4j − 1 for some integer j.
(h) for every odd integer m, m2 = 8k + 1 for some integer k. (Hint: Use the fact that k(k + 1)is an even integer for every integer k.)
(i) for all odd integers m and n, if mn = 4k − 1 for some integer k, then m or n is of the form 4j – 1 for some integer
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