please help 1 Prove using contraposition that for all intege

please help

1- Prove using contraposition that for all integers m and n, if mn is even, then m is even or n is even.

2- Prove that for all integers a, b, c, s, and t, if a|b and a|c, then a|(sb ? tc).

3- Prove that for any even integer m and odd integer n that 3m + 5n is odd.

Let us assume if possible mn is even when m and n are not even.

mn is even means

mn = 2k for some integer k

This implies m*n = 2*k

Hence 2 must be a factor of either m or n as k is an integer.

Hence contradiciton

mn is even implies either m or n is even.

-----------------------------------------------

2) a divides b and a divides c. This implies a divides sb also and a divides tc also

When a divides sb, sb = ka, similarly tc = ma

sb-tc = (k-m)a where k-m is again integer.

Hence a/sb-tc

3) even integer m, 3m is even.

For odd integer n, 5n is odd as both factors do not have 2.

3m+5n = even+odd is again odd.