15. The 5 -digit number ' $X X 4 X Y$ ' is exactly divisible by 165 . What is the value of $X+Y$ ?
15. 14 First note that $165=3 \times 5 \times 11$. Hence, for ' $X X 4 X Y$ ' to be exactly divisible by 165 , it must be exactly divisible by 3,5 and 11 . A number is divisible by 3 if and only if the sum of its digits is divisible by 3 so $3 X+4+Y$ is divisible by 3 and hence $4+Y$ is divisible by 3. A number is divisible by 5 if and only if its last digit is 5 or 0 so $Y=5$ or 0 . Since $4+Y$ is divisible by 3 then $Y=5$. A number is divisible by 11 if and only if the sum of its digits with alternating signs is divisible by 11 so $X-X+4-X+Y$ is divisible by 11. Hence $9-X$ is divisible by 11 and so $X=9$. Hence the value of $X+Y$ is $9+5=14$.