HCF and LCM from prime factors

Two words to recall

• The HCF (highest common factor) of some numbers is the largest number that divides each of them.
• The LCM (lowest common multiple) is the smallest number that each of them divides.

For small numbers you can list factors and multiples. For larger numbers that is slow, and it is easy to miss one. The prime factorisation gives both at once.

The HCF and the LCM from the primes

Write each number as a product of prime powers.

• HCF: take the primes that are in every number, each with its smallest power.
• LCM: take every prime that appears, each with its greatest power.

A worked example. Find the HCF and the LCM of $84$ and $120$.

1. Factorise. $84 = 2^2 \times 3 \times 7$ and $120 = 2^3 \times 3 \times 5$.
2. HCF. The common primes are $2$ and $3$. Smallest powers: $2^2$ and $3^1$. HCF $= 4 \times 3 = 12$.
3. LCM. All the primes: $2, 3, 5, 7$. Greatest powers: $2^3, 3^1, 5^1, 7^1$. LCM $= 8 \times 3 \times 5 \times 7 = 840$.

The picture shows why. The overlap holds the primes that both numbers share: that is the HCF. Everything in the two circles together is the LCM.

Three numbers

The rule works for any number of numbers. Take $12 = 2^2 \times 3$, $18 = 2 \times 3^2$ and $30 = 2 \times 3 \times 5$.

• HCF: $2$ and $3$ are in all three, with smallest powers $2^1$ and $3^1$. HCF $= 6$.
• LCM: greatest powers $2^2, 3^2, 5^1$. LCM $= 4 \times 9 \times 5 = 180$.

A rule for two numbers

For $84$ and $120$: HCF $\times$ LCM $= 12 \times 840 = 10\,080$, and $84 \times 120 = 10\,080$. The same!

For two numbers $a$ and $b$, this is always true:

The reason is in the powers. For each prime, the HCF takes the smaller power and the LCM the greater one. Together they use both powers, exactly as the product $a \times b$ does.

Careful: only for two numbers. For $4$, $6$ and $10$, the HCF is $2$ and the LCM is $60$. So HCF $\times$ LCM $= 120$, but $4 \times 6 \times 10 = 240$.

The rule has two uses. It finds the LCM quickly when you know the HCF. It also checks your answers.

Two facts follow. The HCF always divides the LCM. And if two numbers have no common prime, their HCF is $1$ and their LCM is their product. Such numbers are called co-prime.

Practise with new numbers

Try it in Python

The program factorises both numbers, then builds the HCF from the smaller powers and the LCM from the greater ones.

Common mistakes

• Swapping the powers. The HCF takes the smallest power, the LCM the greatest. A quick check: the HCF can never be larger than the smallest number.
• Leaving out a prime from the LCM. A prime that is in only one number still goes into the LCM. Only the HCF needs the prime in every number.
• Using the product rule for three numbers. HCF $\times$ LCM $=$ product holds for two numbers only.
• Multiplying the numbers to get the LCM. $6 \times 8 = 48$ is a common multiple of $6$ and $8$, but the LCM is $24$.

Remember these

• HCF: common primes, smallest powers. LCM: all primes, greatest powers.
• For two numbers, HCF $\times$ LCM $= a \times b$. Not for three.
• The HCF divides the LCM. Co-prime numbers have HCF $1$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Find the HCF and the LCM of $2^4 \times 3$ and $2^2 \times 3^2 \times 5$.

2. What are the HCF and the LCM of the primes $17$ and $23$?

3. Two numbers have an HCF of $9$ and an LCM of $270$. One of them is $54$. Find the other.

4. Why can the LCM of two numbers never be smaller than either of them?

5. Is it true that HCF $\times$ LCM $= 2 \times 3 \times 4$ for the numbers $2$, $3$ and $4$?