Exercise 2.2.4 (First examples, part 4)

Question

The fact that the product of any three consecutive integers is divisible by $3$ leads to the identical congruence $x(x+1)(x+2) \equiv 0 \pmod 3$. Generalise this, and write an identical congruence modulo $m$.

Richard Ganaye · Accepted Answer

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

ewcommand{\N}{\mathbb{N}}

\begin{proof}
Since the product of any $m$ consecutive integers is divisible by $m$, we obtain, for all $x \in \Z$,
$$x(x+1)\cdots(x+m-1) \equiv 0 \pmod m.$$
We can also write
$$x(x-1)\cdots(x-(m-1)) \equiv 0 \pmod m.$$
\end{proof}

Exercise 2.2.4 (First examples, part 4)

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Exercise 2.2.4 (First examples, part 4)

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