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

Theorem

Let $n$ be an integer.

Then:

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

where $\divides$ indicates divisibility.

Proof

Suppose $n$ is even.

Then $2 \divides n$ by definition.

Hence from Divisor Divides Multiple:

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

$\Box$

Suppose $n$ is odd.

Then $n + 1$ is even

Then $2 \divides n + 1$ by definition.

Hence from Divisor Divides Multiple:

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

$\blacksquare$

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 $2.2$: $5 \ \text {(a)}$