Least Common Multiple Formula with Examples

Key Points

• The least common multiple formula is $\text{LCM}(a,b) = \frac{a \times b}{\text{HCF}(a,b)}$, dividing the product by the highest common factor.

• By prime factorisation, the LCM takes the highest power of every prime appearing in either number.

• Multiplying the two numbers gives the LCM only when they share no factor, so the shortcut works for coprime pairs and fails otherwise.

• The product relationship holds for two numbers but not for three or more, where the primes must be compared directly.

• The LCM is never smaller than the larger number and never larger than the product, which bounds every answer.

What Is the Least Common Multiple Formula?

The least common multiple of two numbers is the smallest positive number both divide into. For two numbers it can be computed directly:

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

The alternative route uses prime factorisation, taking the highest power of each prime that appears. Euclid's Elements is the classical home of this material, and MacTutor notes that its books seven to nine deal with number theory.

Why Dividing by the HCF Works

Multiplying $a$ by $b$ certainly produces a common multiple, but usually not the smallest one. The product counts every shared factor twice, once from each number.

Take 12 and 18. Their product is 216, but both contain a factor of 6, so 216 carries that 6 twice over. Dividing by the highest common factor removes exactly one of the duplicates.

That is why the formula divides rather than subtracts. The HCF is precisely the overlap between the two factorisations, so removing it once leaves each prime at its highest power and nothing more.

Variable Key

Examples of the Least Common Multiple Formula

Example 1

Find the LCM of 12 and 18.

The HCF of 12 and 18 is 6, so

$$\text{LCM} = \frac{12 \times 18}{6} = \frac{216}{6} = 36$$

Final answer: 36

Example 2

Find the LCM of 8 and 12 by multiplying them together.

The product is $8 \times 12 = 96$, which both numbers do divide into.

But 96 is not the smallest such number. Both 8 and 12 divide into 24, which is far smaller, so 96 is a common multiple rather than the least one.

The two numbers share a factor of 4, which the product counts twice. Dividing it out once:

$$\text{LCM} = \frac{8 \times 12}{4} = \frac{96}{4} = 24$$

Final answer: 24. Multiplying works only when the numbers share no factor at all.

Example 3

Find the LCM of 7 and 9.

These share no common factor, so their HCF is 1:

$$\text{LCM} = \frac{7 \times 9}{1} = 63$$

Final answer: 63. For coprime numbers the LCM really is the product, which is the one case the shortcut is valid.

Example 4

Find the LCM of 12, 18 and 30 by prime factorisation.

Factorise each, then take the highest power of every prime:

$$12 = 2^2 \times 3, \qquad 18 = 2 \times 3^2, \qquad 30 = 2 \times 3 \times 5$$ $$\text{LCM} = 2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180$$

Final answer: 180

The product-over-HCF formula does not extend to three numbers, so prime factorisation is the reliable method here.

Example 5

Two buses leave a depot together, one every 18 minutes and one every 24 minutes. When do they next leave together?

The HCF of 18 and 24 is 6:

$$\text{LCM} = \frac{18 \times 24}{6} = \frac{432}{6} = 72$$

Final answer: after 72 minutes, so 1 hour and 12 minutes.

This is the characteristic use of an LCM: the first moment two repeating cycles coincide again.

Don't Confuse the LCM With These Three

LCM and HCF point in opposite directions, which is the quickest way to check an answer. An LCM smaller than either number, or an HCF larger than either, signals that the two have been swapped.

Where Do Students Trip Up on the LCM?

Mistake 1: Multiplying the two numbers

Where it slips in: The product is always a common multiple, so it passes a quick check and looks finished.

Don't do this: $\text{LCM}(8,12) = 96$

The correct way: Divide by the HCF, giving 24. The product only equals the LCM when the HCF is 1.

Students who check whether the numbers share a factor before doing anything know immediately whether the shortcut is available. If they do share one, the product is guaranteed to overshoot.

Mistake 2: Taking the lowest power of each prime

Where it slips in: The word "least" in the name suggests choosing the smaller exponents.

Don't do this: $\text{LCM}(12,18) = 2 \times 3 = 6$

The correct way: Take the highest power of each prime, giving $2^2 \times 3^2 = 36$. Lowest powers give the HCF, not the LCM.

Mistake 3: Using the product formula on three numbers

Where it slips in: The two-number formula is memorable, so it gets extended by analogy.

Don't do this: $\frac{12 \times 18 \times 30}{\text{HCF}(12,18,30)}$

The correct way: Use prime factorisation for three or more numbers, as in Example 4. The product relationship is specific to pairs.

Key Takeaways

• The least common multiple formula is $\text{LCM}(a,b) = \frac{a \times b}{\text{HCF}(a,b)}$.

• Dividing by the HCF removes the shared factors the product counts twice.

• By prime factorisation, take the highest power of each prime; lowest powers give the HCF instead.

• The pair formula does not extend to three numbers, where prime factorisation is the reliable route.

• To work through number theory with a teacher, explore a 6th grade math tutor or math classes online.

Practice These Three Before Moving On

Find the LCM of 15 and 20, then of 9 and 16. Finally find the LCM of 6, 10 and 15 by prime factorisation, and check each answer is at least as large as the biggest number given.

Want a live Bhanzu trainer to walk through more number theory problems? Book a free demo class.

Read More

• Prime Numbers

• What Is a Composite Number

• What Is a Multiple

• Factoring Methods

• How to Do Long Division