# Find the unknown digits X and Y of the five-digit number 313xy if it is divisible by 36. Of all such numbers

**Find the unknown digits X and Y of the five-digit number 313xy if it is divisible by 36. Of all such numbers in the answer, indicate the one with the greatest product of digits.**

If 313xy is divisible by 36, then it must be divisible by all divisors of 36.

36 = 2 * 18 = 4 * 9 = 6 * 6 = 3 * 12.

The number 313xy is divisible by an integer by 2, 3, 4, 6, 9, 12, 18.

3 * 1 * 3 * x * y is the greatest, so x and y are not equal to 0.

313xy is divisible by 2, so y can be 0, 2, 4, 6, 8.

if y = 0, then 3 * 1 * 3 * x * 0 = 0, is not the answer.

313xy is divisible by 9, so 3 + 1 + 3 + x + y = 7 + x + y is divisible by 9.

The numbers 9, 18, 27 are divisible by 9 …

y = 2, 4, 6, 8

7 + x + y = 9

x + y = 2, if y = 2, then x = 0, is not the answer.

7 + x + y = 18.

x + y = 11, x = 11 – y.

If y = 2, x = 11 – 2 = 9. The number is 31392.

y = 4, x = 11 – 4 = 7, Number 31374.

y = 6, x = 11 – 6 = 5, Number 31356.

y = 8, x = 11 – 8 = 3, Number 31338.

7 + x + y = 27

x + y = 20 is not a solution, since x <10, y <10.

The number 313xy is divisible by 4, so xy is divisible by 4.

92: 4 = 23.

74: 4 = 18, remainder 2.

56: 4 = 14.

38: 4 = 9, remainder 2.

The resulting numbers are 31392 and 31356.

31392: 36 = 872, 3 * 1 * 3 * 9 * 2 = 162.

31356: 36 = 871, 3 * 1 * 3 * 5 * 6 = 270.

Answer: 31356.