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:

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To which I had two questions, the second of greater import:

  1. 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$3$?
  2. 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.

  • $\begingroup$ I'm sure this notation is defined earlier in your textbook. $\endgroup$ – Ted Feb 2 at 18:49
  • $\begingroup$ The notation $a\equiv b\pmod n$ is equivalent to saying that $n$ divides $a-b$; in other words, $a$ and $b$ are congruent modulo $n$, ie, they represent the same equivalence class in the set of integers modulo $n$ (denoted by $\Bbb Z/n\Bbb Z$) $\endgroup$ – learner Feb 2 at 18:54
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    $\begingroup$ I recommend that you start with reading about the basics of modular arithmetic and properties on modular equivalence, for example $a\equiv b\pmod n$ implies $a^m\equiv b^m\pmod n$, etc $\endgroup$ – learner Feb 2 at 18:56
  • $\begingroup$ Have you yet studied modular arithmetic or congruences? $\endgroup$ – Bill Dubuque Feb 2 at 19:02
  1. 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 $1 \pmod 3$.

  2. 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 $1 \pmod 3.$ $0^2 = 0 \times 0 = 0.$


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