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

if direction

assume (x+9y) is divisible by 19.

x+9y = 19k

9x+5y = 9(19k-9y)+5y =

19(9k) - 76y =

19(9k - 4y)

and thus we see (9x+5y) is divisible by 19

you try the only if direction

thanks