Exercise 205Number theory › Divisibility and gcdHigh schoolAdded 2 Nov 2026

Consecutive integers are coprime

#coprime#consecutive integers#Euclidean algorithm

Statement

Show that for every integer $n$ , $\divisibility(n,n+1)=1$ .

Hint

Any common divisor divides their difference.

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Let $d=\divisibility(n,n+1)$ . Then $d\mid n$ and $d\mid n+1$ , so $d\mid (n+1)-n=1$ . Hence $d=1$ .

Remark. This is the simplest example of the Euclidean algorithm: $n+1=1\cdot n+1$ and $n=1\cdot n+0$ , so the last nonzero remainder is $1$ .