Exercise 1.2.20 (Integers of the form $3k + r$)

Question

Given integers $a$ and $b$, a number $n$ is said to be of the form $ak+b$ if there is an integer $k$ such that $ak+b =n$. Thus the numbers of the form $3k+1$ are $-8,-5,-2,1,4,7,10,\ldots$. Prove that every integer is of the form $3k$ or of the form $3k+1$ or of the form $3k+2$.
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Richard Ganaye · Accepted Answer

ewcommand{\Z}{\mathbb{Z}}

begin{proof} The division algorithm (Theorem 1.2) shows that, for any integer $n \in \Z$, there exist unique integers $k$ and $r$ such that
$$n = 3 k + r,\qquad 0 \leq r < 3.$$
So every integer is of the form $3k$ or of the form $3k+1$ or of the form $3k+2$, the three cases being mutually exclusive.
\end{proof}

Exercise 1.2.20 (Integers of the form $3k + r$)

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Exercise 1.2.20 (Integers of the form $3k + r$)

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