Consecutive Integers are Coprime/Proof 3
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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 $2.2$: $12$