Lowest Common Multiple of Integers/Examples/n and n+1

Example of Lowest Common Multiple of Integers

Let $n \in \Z_{>0}$ be a strictly positive integer.

The lowest common multiple of $n$ and $n + 1$ is:

$\lcm \set {n, n + 1} = n \paren {n + 1}$

Proof

We find the greatest common divisor of $n$ and $n + 1$ using the Euclidean Algorithm:

\(\text {(1)}: \quad\)

\(\ds n + 1\)

\(=\)

\(\ds 1 \times n + 1\)

\(\text {(2)}: \quad\)

\(\ds n\)

\(=\)

\(\ds n \times 1\)

Thus $\gcd \set {n, n + 1} = 1$.

Hence by definition $n$ and $n + 1$ are coprime.

The result follows from LCM equals Product iff Coprime.

$\blacksquare$

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-2}$ Divisibility: Exercise $5 \ \text {(e)}$