Consecutive Integers are Coprime/Proof 3

Theorem

$\forall h \in \Z$, $h$ and $h + 1$ have only two common factors: $1$ and $-1$.

That is, consecutive integers are always coprime.


Proof

A direct application of GCD of Integer with Integer + $n$:

$\gcd \set {a, a + n} \divides n$

$\blacksquare$


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 $2.2$: $12$
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