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

Mellie Jul 30, 2015

#2**+11 **

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.

Melody Jul 31, 2015

#3**+5 **

"*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.

Alan Aug 1, 2015