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 me prove the "only if" part first.
x+9y≡0(mod19)9(x+9y)≡0(mod19)9x+81y≡0(mod19)9x+19(4y)+5y≡0(mod19)9x+5y≡0(mod19)
The "if" part would just be the above, but writing the steps in reverse order.
