SOLUTION: For each integer n, its square can be written uniquely as {{{ n^2=10k+r }}}, where r is an integer in the range 0 ≤ r ≤ 9. Determine which values of r can occur and whi

That's the same as asking

"What digits can squares of integers end in?"

and

"What digits can squares of integers NOT end in?"

The last digit of the product of any two
integers is always the same digit as the
last digit product of their last two 
digits.  So we look at these products:

0×0 = 0
1×1 = 1
2×2 = 4
3×3 = 9
4×4 = 16
5×5 = 25
6×6 = 36
7×7 = 49
8×8 = 64
9×9 = 81

Those all end with 0,1,4,6,or 9

So r can be 0,1,4,6, or 9

So r cannot be 2,3,5,7, or 8

Edwin