Let x and y be integers. Show that 9x + 5y is divisible by 19 if and only if x + 9y is divisible by 19.
The only numbers that work are of the form x = 19n + 9 and y = 19n + 18. Then 9x + 5y = 9(19n + 9) + 5(19n + 18) = 19(4n + 9) x + 9y = 19n + 9 + 9(19n + 18) = 19(10n + 9)
The result follows.
