Divisor of Integer/Examples/6 divides n (n+1) (n+2)/Proof 1

Theorem

Let $n$ be an integer.

Then:

$6 \divides n \paren {n + 1} \paren {n + 2}$

Proof

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

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

From $2$ divides $n \paren {n + 1}$:

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

and so:

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

Hence:

$2 \times 3 = 6 \divides \paren {n + 1} \paren {n + 2}$

$\blacksquare$