If the whole number multiples of 36 that are both more than 36 and less than 36 squared, how many are squares of whole numbers?

This is different than the usual questions we get.

We're given a set of some multiples of 36:

We're asked how many of those are perfect squares.

36 is of course already a perfect square, so for the product to be a perfect square the other factor must be as well. So this is equivalent to:

How many perfect squares are there between 2 and 35?

Just listing them, there's

4 squares

There are 36-2=34 whole number multiples of 36 between 36 and 36^2=1296, i.e. k*36 where k=2,3,4,35.

Since 36 is itself a perfect square, we only have to look at k to make sure k is a perfect square. Between 2 and 35, perfect squares are 4,9,16,25, so when k ∈ {4,9,16,25}, k*36 is a perfect square.

Therefore there are four multiples of 36 strictly between 36 and 1296 that are perfect squares, or squares of whole numbers.