Square numbers are the result of multiplying the same two numbers together, and the last digit of a square number is dependent on the last digit of the number that you are squaring. There are only 10 numbers to test.
\(0^2=0\)
\(1^2=1\)
\(2^2=4\)
\(3^2=9\)
\(4^2=16\)
\(5^2=25\)
\(6^2=36\)
\(7^2=49\)
\(8^2=64\)
\(9^2=81\)
As can be seen, the last digit of a square number must be 0, 1, 4, 5, 6 or 9. Therefore no square number can have a last digit of 2, 3, 7 or 8.