LCM Shortcut for Two Numbers

Trick / Shortcut

Topic

Divisibility

How It Works

The Least Common Multiple (LCM) of two numbers $a$ and $b$ can be found using their Highest Common Factor (HCF):

$\text{LCM}(a, b) = \frac{a \times b}{\text{HCF}(a, b)}$

To find the HCF of two numbers, repeatedly replace the larger number with the remainder after division until the remainder is 0.
The last non-zero remainder in the sequence of divisions is the HCF.

Example

Let's find the LCM of 56 and 42.

First, find the HCF of 56 and 42 using the Euclidean Algorithm: $56 = 42 \times 1 + 14$ $42 = 14 \times 3 + 0$

The last non-zero remainder is 14, so $\text{HCF}(56, 42) = \textbf{14}$.

Now, use the LCM formula: $\text{LCM}(56, 42) = \frac{56 \times 42}{14} = \frac{2352}{14} = \textbf{168}$