Is n^2 congruent to 0 or 1 (mod 3) for any integer n?

[phyguy321]

proof:
n^2 congruent 0 or 1 (mod3) for any integer n

 

[jeffreydk]

Try considering the two cases where n^2is either even or odd and how that relates to the congruence modulo 3.

 

[snipez90]

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.

 

[NoMoreExams]

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?