Let $x$ and $y$ be integers. Show that $9x + 5y$ is divisible by 19 if and only if $x + 9y$ is divisible by 19.

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I really like this solution Alan, thank you.

For the last bit you just have to show that 9m/19 is not an integer for  0<m<19

can't you just say that 9 and 19 are relatively prime so 9m/19 will only be an integer if m is a multiple of 19.  Since m cannot be a multiple of 19, 9m/19 is not an integer.

That is, 9m is not divisable by 19.

"For the last bit you just have to show that 9m/19 is not an integer for  0<m<19 can't you just say that 9 and 19 are relatively prime so 9m/19 will only be an integer if m is a multiple of 19.  Since m cannot be a multiple of 19, 9m/19 is not an integer. That is, 9m is not divisable by 19"

Yes.  Much neater Melody.

Thankyou Alan