Least common multiple

Definitions

  • The least common multiple (LCM or l.c.m.) or two (or more) integers it the smallest integer that can be divided by them.
  • There is an infinite number of common multiples, but we are most interested in the smallest of them.
  • The LCM of coprime numbers is simply their product.
  • The product of two (or more) numbers is a common multiple, but it is not necessarily the least common multiple.

Algorithm

  1. We factorize each number into its primes;
  2. We take every prime that appears; and
  3. We take them with the highest power.
  4. The LCM is the product of those primes raised to their highest powers.
Example: Finding LCM of two numbers

To find the LCM of \(48\) and \(90\), we express each of them as the product of primes \begin{align} 48&=2^4\times 3 \\ 18&=2\times 3^2\times 5 \end{align}

We take \(2\) to its highest power (which is \(4\)), \(3\) to its highest power (which is \(2\)), and \(5\) to its highest power (which is \(1\)). Therefore, \[\operatorname{LCM}(48,90)=2^4\times 3^2\times 5=720\]

Important Note

Although we may be able to guess the LCM for small numbers relatively easily, a systematic approach is necessary when working with larger numbers or a lot of numbers.

Example: Finding LCM of three numbers

To find the LCM of \(180\), \(150\) and \(120\), we express each of them as the product of primes \begin{align} 180 &= 2^2\times 3^2\times 5 \\ 150 &= 2\times 3\times 5^2 \\ 120 &= 2^3\times 3\times 5 \end{align}

We take \(2\) to its highest power (which is \(3\)), \(3\) to its highest power (which is \(2\)), and \(5\) to its highest power (which is \(2\)). Therefore, \[\operatorname{LCM}(180,150,120)=2^3\times 3^2\times 5=360\]

LCM for symbolic expressions

The same principles are followed if we want to get the LCM of two (or more) expressions with unknowns (i.e., placeholder letters) in them. For example, the LCM of \(6x^2y^3\) and \(9x^3yz^2\) is obtained by factorizing their numeric coefficients and expressing them as products. So, \begin{align} 6x^2y^3&=2\times 3\times x^2\times y^3 \\ 9x^3yz^2&=3^2\times x^3\times y\times z^2 \end{align} Hence, the LCM is \(2\times 3^2\times x^3\times y^3\times z^2=18x^3y^3z^2\).

Uses of the LCM

It is typically used in finding the common denominator of rational numbers or expressions, in addition to several other important uses. For example to compare \(\frac{2}{15}\) and \(\frac{4}{21}\), we have to find a common denominator for them. This common denominator is the LCM of \(15\) and \(21\), so we factorize them to find that \begin{align}\require{cancel} 15&=3\times 5 \\ 21&=3\times 7 \end{align}

The LCM of those numbers is \(3\times 5\times 7\). We need to make the denominator of both rational numbers that LCM, so \begin{align} \frac{2}{15}&=\frac{2}{3\times 5}=\frac{2\times 7}{3\times 5\times 7}=\frac{14}{168} \\ \frac{4}{21}&=\frac{4}{3\times 7}=\frac{4\times 5}{3\times 5\times 7}=\frac{20}{168} \end{align}

Therefore, \(\frac{4}{21}\gt\frac{2}{15}\) because \(\frac{14}{168}\gt\frac{20}{168}\).