Prove that (a) there exist integers m and n such that 2m+7n=1. (b) there exist integers...

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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 rands such that 3r+ 8s=f. (6) 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.) (1) for all odd integers m and n, if mn= 4k-1 for some integer k, then m or n is of form 4j-1 for some integer 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 rands such that 3r+ 8s=f. (6) 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.) (1) for all odd integers m and n, if mn= 4k-1 for some integer k, then m or n is of form 4j-1 for some integer