N2 has the form 3k or 3k+1 for some integer k

[annoymage]

Homework Statement

n² has the form 3k or 3k+1 for some integer k

Homework Equations

n/a

The Attempt at a Solution

i've tried to table it for me to see, but still i don't have the general idea, should i show n² is either divisable by 3k or 3k+1? if yes i still don't know owho

 

[HallsofIvy]

Any integer, divided by 3, has remainder 0, 1, or 2, thus any number can be written as "3k" (remainder 0), "3k+ 1" (remainder 1), or "3k+ 2" (remainder 2).

Now try squaring those:

$(3k)^2= 9k^2= 3(3k^2)$, "3 times an integer" and so is of the first kind above.

$(3k+ 1)^2= 9k^2+ 6k+ 1= 3(3k^2+ 2k)+ 1$, "3 times an integer plus 1" and so of the second kind above.

$(3k+ 2)^2= 9k^2+ 12k+ 4$$= 9k^2+ 12k+ 3+ 1= 3(3k^2+ 4k+ 1)+ 1$ which is again of the form "3 times an integer plus 1".

Note that none of those three forms, 3k, 3k+1, and 3k+ 2 (and every integer can be written in one of those forms), have a square of the form 3n+ 2.