Question

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?

Answer 1

This is different than the usual questions we get.

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

$\{ 2\times 36, 3 \times 36, 4 \times 36, ..., 35 \times 36 \}$

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 $2^2, 3^2, 4^2 \textrm{ and } 5^2$

Answer: 4 squares

Answer 2

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.