Divisor of Integer/Examples/3 divides n(n+1)(n+2)

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Theorem

Let $n$ be an integer.

Then:

$3 \divides n \paren {n + 1} \paren {n + 2}$

where $\divides$ indicates divisibility.


Proof

$n$ is of one of these forms:

\(\ds n\) \(=\) \(\ds 3 k\)
\(\ds n\) \(=\) \(\ds 3 k + 1\)
\(\ds n\) \(=\) \(\ds 3 k + 2\)

for some $k \in \Z$.


Suppose $n = 3 k$.

Then $3 \divides n$ by definition.


Suppose $n = 3 k + 1$.

Then:

$n + 2 = 3 k + 3 = 3 \paren {k + 1}$

Thus:

$3 \divides n + 2$.


Suppose $n = 3 k + 2$.

Then:

$n + 1 = 3 k + 3 = 3 \paren {k + 1}$

Thus:

$3 \divides n + 1$.


Hence from Divisor Divides Multiple:

$3 \divides n \paren {n + 1} \paren {n + 2}$

$\blacksquare$


Sources