Try to be systematic. How can you narrow down the cases to be tested?
If the last digit of a square number is 9 what can you say about the number?
Can you write down an expression for a number using its digits and powers of 10?
Note: Other problems using modulus arithmetic: Dirisibly Yours , Obviously, Rational Round
An article that explains the basic ideas of modulus arithmetic: Modulus Arithmetic and Solution to Dirisibly Yours.