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
- 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $2$: Divisibility Theory in the Integers: $2.2$ The Greatest Common Divisor: Problems $2.2$: $5 \ \text {(a)}$