A positive two-digit number is even and is a multiple of 11. The product of its digits is a perfect cube. What is this two-digit number?
I know that any multiple of 11 must have the same sum of its odd and even digits. So, if the two-digit number is even and a multiple of 11, it must be of the form 11k, where k is an even number from 2 to 9. The possible values for 11k are 22, 44, 66, and 88. Now, I need to check which of these values have a product of digits that is a perfect cube. The product of the digits of 22 is 2 * 2 = 4, which is not a perfect cube. The product of the digits of 44 is 4 * 4 = 16, which is not a perfect cube. The product of the digits of 66 is 6 * 6 = 36, which is not a perfect cube. The product of the digits of 88 is 8 * 8 = 64, which is a perfect cube, since 64 = 4 * 4 * 4 = 4^3. Therefore, the only possible two-digit number that satisfies the given conditions is 88. 