Exercises: Simple fractions

We try the square numbers one by one. Since ${10}^2=100$ is already a three-digit number, we should not get too many two-digit square numbers:

$\begin{array}{rll}{1}^2& =1& |\text{ not two-digit}\\ 2^2& =4& |\text{ not two-digit}\\ 3^2& =9& |\text{ not two-digit}\\ 4^2& =16& |\text{ even square number}\\ 5^2& =25& |\text{ 1st odd square number}\\ 6^2& =36& |\text{ even square number}\\ 7^2& =49& |\text{ 2nd odd square number}\\ 8^2& =64& |\text{ even square number}\\ 9^2& =81& |\text{ 3rd odd square number}\\ {10}^2& =100& |\text{ already three-digit}\end{array}$

All square numbers from 100 on are no longer two-digit numbers. Thus there are 3 square numbers, whose final digit is odd, namely 25, 49 and 81.