LCM Using Prime Factorisation

Great progress so far in Finding the Lowest Common Multiple! We are already at lesson three of five in this course. Over the first two lessons, you explored common multiples and then learned how to find the LCM by listing multiples until the first shared value appeared. That listing method is clear and reliable, but it can become slow when numbers get larger. Today, you will learn a more efficient approach: finding the LCM using prime factorisation.

Consider trying to find $\text{LCM}(48, 72)$ by listing. You might need to write out many multiples of each number before spotting the first match. For even bigger numbers, the lists grow quickly and the process becomes tedious.

Prime factorisation gives us a shortcut that works directly with the structure of the numbers rather than scanning long lists. As you may recall from earlier in this learning path, every whole number greater than $1$ can be written as a unique product of prime factors. That breakdown is exactly the tool we will use now.

Here is the key insight: for a number to be a multiple of $12$, it must contain at least the same prime factors that $12$ has, each raised to at least the same power. Since $12 = 2^2 \times 3$, any multiple of $12$ needs at least two $2$s and at least one $3$ among its prime factors. The same logic holds for any number.

So for a number to be a common multiple of two numbers, it must carry enough prime factors to cover both. The LCM is the smallest such number, which means we take every prime that appears in any of the factorisations, each at its highest power across the numbers — nothing more, nothing less.

Let us find $\text{LCM}(12, 18)$ using prime factorisation. Imagine two rotating displays in a shop window: one completes a cycle every $12$ seconds, the other every $18$ seconds. To find when both displays first reset together, we need the LCM of $12$ and $18$.

Step 1 — Factorise each number.

Step 2 — Identify every prime that appears in any factorisation. The primes involved are $2$ and $3$.

Step 3 — For each prime, choose the highest power across both factorisations.

Step 4 — Multiply the highest powers together.

We can verify: $36 \div 12 = 3$ and $36 \div 18 = 2$. Both results are whole numbers, confirming that $36$ is indeed a common multiple. Because we used only the minimum prime power needed to cover each number, it is the lowest common multiple. Back to our shop-window scenario — both displays first reset together after 36 seconds.