How many perfect numbers are there

Since 10 is the smallest two-digit number and 99 is the largest two-digit number, that means the perfect square would have to be in between 1*0 = 0 and 9*9 = 81. So the perfect square has to be 0, 1, 4, 9, 16, 25, 36, 49, 64, or 81. Find out the factors of these numbers, then only consider the factor pairs where both factors are single digits.

Let's start with 0. The only way to get 0 is if one of the digits is 0, and it can't be the first digit. So 10, 20, 30, and so on up through 90 gives us a product of 0.

Now move on to 1. The only number that does this is 11, since the only factor of 1 is itself.

Now consider 4. The factors are 1*4 and 2*2, so 14, 22, and 41 are the three numbers that give us a product of 4.

Likewise for 9, we get 19, 33 and 91.

The factors of 16 are 1, 2, 4, 8, 16. So 28, 82, and 44 are the only double-digit numbers to make here.

For 25, there's only "55". Likewise for 49 there's only "77", and for 81 we only have "99".

The only factors of 36 that are single digits are 1, 2, 3, 4, 6, 9. From these, we can make 49, 94 and 66. And finally from 64 we only have 1, 2, 4, and 8, giving us only 88.

So in total, there are 9 + 1 + 3 + 3 + 3 + 1 + 1 + 1 + 3 + 1 = 26 two-digit numbers whose product of digits is a perfect square.