Question: prove that for integers x and y there are no solutions to the modular equation $$ 9x + 10\equiv 6y - 1 \mod 15 $$
So my first thought was to assume that the congruence does have solutions, so 15 would divide the difference between the two equations. Meaning: $$ (9x + 10) - (6y - 1) = 15k \\ 9x - 6y + 11 = 15k $$ for some integer k. Then I notice that we have a lot of multiples of 3 here, so I rearrange it like this: $$ 11 = 15k - 9x + 6y \\ 11 = 3(5k - 3x + 2y) $$ This means 11 is an integer multiple of three. Clearly this is false, so we have arrived at a contradiction. Therefore there is no integer k such that $9x - 6y + 11 = 15k$, and thus the original modular equation has no solutions.
Is my logic valid? Is this the best way to prove this, or is there a more succinct and correct way? Thank you!