Least common multiple

The least common multiple of two integers is the smallest positive integer between all the multiples of both. It is usually written as LCM(a, b).^[1] Likewise, the LCM of more than two integers is the smallest positive integer that is divisible by each of them.^[2][3]

Overview

Elementary arithmetic

In elementary arithmetic, the LCM is also the "lowest common denominator" (LCD) that must be calculated, before fractions can be added, subtracted or compared.

A multiple of a number is the product of that number and an integer. For example, 10 is a multiple of 5 because 5 × 2 = 10, so 10 is divisible by 5 and 2. Because 10 is the smallest positive integer that is divisible by both 5 and 2, it is the least common multiple of 5 and 2. By the same principle, 10 is the least common multiple of 5 and 2 as well.

Relations with the greatest common divisor

It is known that:

${\displaystyle \operatorname {GCD} (a,b)\cdot \operatorname {LCM} (a,b)=|a\cdot b|}$

where ${\displaystyle \operatorname {GCD} (a,b)}$ is the greatest common divisor of a and b, This formula is often used to compute the LCD, by first finding the GCD of a and b.

Related pages

• Scalar
• Algebra
• Arithmetic
• Pure mathematics
• Mathematical analysis
• Greatest common divisor (GCD)