What does HCF stand for?

Highest Common Factor - the largest number that divides both numbers

What does LCM stand for?

Lowest Common Multiple - the smallest number that is a multiple of both

How do you find HCF using a Venn diagram of prime factors?

Multiply the numbers in the centre (the common prime factors)

How do you find LCM using a Venn diagram of prime factors?

Multiply all the numbers in the diagram (centre and regions)

For HCF, do you use the lowest or highest power of each common prime?

Lowest power (the one that divides both)

For LCM, do you use the lowest or highest power of each prime?

Highest power (from either number)

What is the HCF of 24 and 60?

12. 24=2^3 x 3, 60=2^2 x 3 x 5. HCF = 2^2 x 3 = 12

What is the LCM of 42 and 90?

630. 42=2x3x7, 90=2x3x3x5. LCM = 2x3x7x3x5 = 630

When is HCF equal to 1?

When the two numbers have no common prime factors (coprime)

HCF & LCM | AQA GCSE Maths

HCF & LCM

Summary

The Highest Common Factor (HCF) is the largest number that divides both numbers. The Lowest Common Multiple (LCM) is the smallest number that is a multiple of both. Use factor lists, Venn diagrams with prime factors, or highest powers of primes.

1. Highest Common Factor (HCF)

A common factor divides both numbers with no remainder. 1 is always a common factor.

The HCF is the largest common factor. Useful for simplifying fractions and factorising.

Method 1: List factors of each number, find the largest in both lists.

Method 2: Write as product of prime factors, use Venn diagram. HCF = product of numbers in the centre (common primes).

Method 3: Use highest power of each common prime. HCF of 24 and 60: 24=2^3 x 3, 60=2^2 x 3 x 5; HCF = 2^2 x 3 = 12.

Note: HCF can equal one of the numbers (e.g. HCF of 4 and 12 is 4).

2. Lowest Common Multiple (LCM)

A common multiple appears in both times tables. The product of the two numbers is always a common multiple (but not necessarily the lowest).

The LCM is the smallest common multiple. Useful for common denominators and adding fractions.

Method 1: List multiples of each number, find the smallest in both lists.

Method 2: Venn diagram with prime factors. LCM = product of ALL numbers in the diagram.

Method 3: Use highest power of each prime that appears in either number. LCM of 72 and 540: 72=2^3 x 3^2, 540=2^2 x 3^3 x 5; LCM = 2^3 x 3^3 x 5 = 1080.

Note: LCM can equal one of the numbers (e.g. LCM of 4 and 12 is 12).

3. Venn Diagram Method

Write each number as product of prime factors (e.g. 42=2x3x7, 90=2x3x3x5).

Common primes go in the centre. For 42 and 90: put 2 and one 3 in centre.

Remaining primes go in their regions: 7 for 42, 3 and 5 for 90.

HCF = product of centre = 2 x 3 = 6.

LCM = product of all numbers in diagram = 7 x 2 x 3 x 3 x 5 = 630.

4. Worked Example: HCF of 36 and 120

36 = 2^2 x 3^2, 120 = 2^3 x 3 x 5.

Venn diagram: centre has 2, 2, 3 (common primes). HCF = 2 x 2 x 3 = 12.

Alternatively: highest common powers 2^2 and 3^1, so HCF = 4 x 3 = 12.

5. Worked Example: LCM of 36 and 120

6. Key Differences

HCF & LCM

Summary

1. Highest Common Factor (HCF)

A common factor divides both numbers with no remainder. 1 is always a common factor.

The HCF is the largest common factor. Useful for simplifying fractions and factorising.

Method 1: List factors of each number, find the largest in both lists.

Method 2: Write as product of prime factors, use Venn diagram. HCF = product of numbers in the centre (common primes).

Method 3: Use highest power of each common prime. HCF of 24 and 60: 24=2^3 x 3, 60=2^2 x 3 x 5; HCF = 2^2 x 3 = 12.

Note: HCF can equal one of the numbers (e.g. HCF of 4 and 12 is 4).

2. Lowest Common Multiple (LCM)

A common multiple appears in both times tables. The product of the two numbers is always a common multiple (but not necessarily the lowest).

The LCM is the smallest common multiple. Useful for common denominators and adding fractions.

Method 1: List multiples of each number, find the smallest in both lists.

Method 2: Venn diagram with prime factors. LCM = product of ALL numbers in the diagram.

Method 3: Use highest power of each prime that appears in either number. LCM of 72 and 540: 72=2^3 x 3^2, 540=2^2 x 3^3 x 5; LCM = 2^3 x 3^3 x 5 = 1080.

Note: LCM can equal one of the numbers (e.g. LCM of 4 and 12 is 12).

3. Venn Diagram Method

Write each number as product of prime factors (e.g. 42=2x3x7, 90=2x3x3x5).

Common primes go in the centre. For 42 and 90: put 2 and one 3 in centre.

Remaining primes go in their regions: 7 for 42, 3 and 5 for 90.

HCF = product of centre = 2 x 3 = 6.

LCM = product of all numbers in diagram = 7 x 2 x 3 x 3 x 5 = 630.

4. Worked Example: HCF of 36 and 120

36 = 2^2 x 3^2, 120 = 2^3 x 3 x 5.

Venn diagram: centre has 2, 2, 3 (common primes). HCF = 2 x 2 x 3 = 12.

Alternatively: highest common powers 2^2 and 3^1, so HCF = 4 x 3 = 12.

5. Worked Example: LCM of 36 and 120

36 = 2^2 x 3^2, 120 = 2^3 x 3 x 5.

LCM = highest power of each prime: 2^3, 3^2, 5^1. LCM = 8 x 9 x 5 = 360.

Alternatively list multiples: 36, 72, 108, 144, 180, 216, 252, 288, 324, 360; 120, 240, 360. LCM = 360.

6. Key Differences

HCF: use only COMMON primes, take LOWEST power of each.

LCM: use ALL primes from either number, take HIGHEST power of each.

If no common primes, HCF = 1. LCM still uses all primes from both.