06/18/2020 (2011: National Target Round, Problem 3)

Q: When all the factors of 90 are multiplied together, the product can be expressed as 90^k. What is the value of k?

A: 6

Let's list out all the factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90.

We know that the pairs of factors are {1, 90}, {2, 45}, {3, 30}, {5, 18}, {6, 15}, {9, 10}, and that each pair of factors multiplies to 90.

There are 6 pairs, so that means that k = 6.