# The number #90^9# has #1900# different positive integral divisors. How many of these are squares of integers?

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The answer is 250 but I'm not sure how to get there. (Number theory isn't one of my strong suits).

The answer is 250 but I'm not sure how to get there. (Number theory isn't one of my strong suits).

##### 1 Answer

Wow - I get to answer my own question.

#### Explanation:

It turns out that the approach is a combination of combinatorics and number theory. We begin by factoring

The trick here is to figure out how to find squares of integers, which is relatively simple. Squares of integers can be generated in a variety of ways from this factorization:

We can see that

The same reasoning applies to **any combination** of these prime divisors who have even powers also satisfies the conditions. For instance,

Thus the desired number of squares of integers that are divisors of