My text posed the following question (priorly asked on Math StackExchange): Prove that the square of any integer has one of the forms $3k$ or $3k + 1$, $k \in ℤ$, and provided the answer:
To which I had two questions, the second of greater import:
- What informs us to conclude that if $x^2$ is congruent to $0$ or $1$ mod$3$, then $x$ is congruent to $0, 1$ or $2$ mod$3$?
- I do not understand the notation the author provided. What does $x^2 ≡ 0^2$ signify (i.e. without the mod), and how did she obtain that response from $x ≡ 0$ mod$3$ ? Moreover, how does one get from $x^2 ≡ 0^2$ to $x^2 ≡ 0^2 = 0$ mod$3$?
My appreciation in advance.
We are concluding the converse: if $x$ is congruent to $0, 1,$ or $2 \pmod 3$, then $x^2$ is congruent to $0$ or $1 \pmod 3.$ Note that for all $x \in \mathbb Z,$ $x$ is congruent to $0, 1,$ or $2 \pmod 3$. Therefore, by considering these three cases, it is proved that, for all $x \in \mathbb Z,$ $x^2$ is congruent to $0$ or $1 \pmod 3$.
When the text says $x$ is congruent to $0$ or $1 \pmod 3$, that is a short-cut for saying $x$ is congruent to $0 \pmod 3$ or $x$ is congruent to $1 \pmod 3.$ $0^2 = 0 \times 0 = 0.$