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.
