Chapter 1.6, Problem 1E

Prove that
(a) there exist integers m and n such that $\text{2}m+\text{7}n=\text{1}$ .
(b) there exist integers m and n such that $\text{15}m+\text{12}n=\text{3}$ .
(c) there do not exist integers m and n such that $\text{2}m+\text{4}n=\text{7}$ .
(d) there do not exist integers m and n such that $\text{12}m+\text{15}n=\text{1}$ .
(e) for every integer t, if there exist integers m and n such that $\text{15}m+\text{16}n=t$ ,then there exist integers r and s such that $\text{3}r+\text{8}s=t$ .
(f )if there exist integers m and n such that $\text{12}m+\text{15}n=\text{1}$ , then m and n are both positive.
(g) for every odd integer m, if m has the form $\text{4}k+\text{1}$ for some integer k, then $m+\text{2}$ has the form $\text{4}j-\text{1}$ for some integer j.
(h) for every integer m, if m is odd, then $m^{\text{2}}=\text{8}k+\text{1}$ for some integer k.
(i) for all odd integers m and n, if $mn=\text{4}k-\text{1}$ for some integer k, then m or n is of the form $\text{4}j-\text{1}$ for some integer j.