Prove that IF there exist integers m and n such that 12m +15n=1 then m and n are both positive?

Question

This problem can be proved. It's a problem in my book and I have the solution but have trouble understanding it.
solution: suppose 12m+15n=1 for some integer m and n, and suppose further that n and m are not postitive. Then 3 divides the left side but not the right, which is impossible. Therefore, if there exist integers m and n such that 12m +15n =1, then m and n are both positive.

Answer 1

Yes, the if m and n are integers, then 12m + 15n is a multiple of 3 ( i.e. = [3 x (4m +5n)]).

BUT, 1 is not a multiple of 3. therefore, there are NO solutions at all.

The proof is therefore true VACUOUSLY... meaning that the IF part always fails so no contradiction exists on the THEN part...

Answer 2

there doesn't exist any positive integer.
for m=0 or n=0, it is 12 or 15
for m=1 and n=1
it gives 12+15=27
for greater values it wll be greater than 27.

So no positive values will satisfy it.
If you try similar way, you will yourself conclude no negative number satisfy it.

Hence it has no interal solution.