Let $x$ and $y$ be integers. Show that $9x + 5y$ is divisible by 19 if and only if $x + 9y$ is divisible by 19.
$9x + 5y$. There are infinite solutions to your problem, both positive and negative:
($9x95) + ($5x171) =$17,100, which is divisible by 19=$900.
Let x and y be integers.
Show that 9x + 5y is divisible by 19 if and only if x + 9y is divisible by 19.
Let x+9y=n⋅19n∈Z
9x+5y?≡0(mod19)|x=n⋅19−9y9⋅(n⋅19−9y)+5y?≡0(mod19)9n⋅19−81y+5y?≡0(mod19)9n⋅19−76y?≡0(mod19)|76=4⋅199n⋅19−4⋅19y?≡0(mod19)19⋅(9n−4y)≡0(mod19) ✓
+4625 
