Question 632981
<pre>
Every integer ends with either a 0,1,2,3,4,5,6,7,8, or 9.

The last digit of the square of a number that ends with 0, ends with 0×0 or 0

The last digit of the square of a number that ends with 1, ends with 1×1 or 1

The last digit of the square of a number that ends with 2, ends with 2×2 or 4

The last digit of the square of a number that ends with 3, ends with 3×3 or 9

The last digit of the square of a number that ends with 4, ends with what 4×4 or 16 ends with, namely 6.

The last digit of the square of a number that ends with 5, ends with what 5×5 or 25 ends with, namely 5.

The last digit of the square of a number that ends with 6, ends with what 6×6 or 36 ends with, namely 6.

The last digit of the square of a number that ends with 7, ends with what 7×7 or 49 ends with, namely 9.

The last digit of the square of a number that ends with 8, ends with what 8×8 or 64 ends with, namely 4.

The last digit of the square of a number that ends with 9, ends with what 9×9 or 81 ends with, namely 1.

So squares can only end with 0,1,4,5,6, and 9

Therefore squares cannot end with 2,3,7, or 8.

Edwin</pre>