n^2 congruent 0 or 1 (mod3) for any integer n
Try considering the two cases where n^2is either even or odd and how that relates to the congruence modulo 3.
Two other methods:
Consider just the squares of the integers from 1 to 9 modulo 3. Then you could generalize to higher numbers since powers of 10 are congruent to 1 (mod 3).
Or, consider a number in base 3. It can end in 0, 1 or 2. Thus a square in base 3 can only end in 0^2 = 0, 1^2 = 1, or 2^2 = 4 = 1 base 3. So a square in base 3 can only end in 0 or 1, which is equivalent to the square leaving a remainder of 0 or 1 upon division by 3.
Going along with what jeffreydk said, if n is an odd integer, what form does it have? what about if n is an even integer?
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