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.