The Highest Common Factor (HCF) is the largest positive integer that divides two or more numbers exactly – so without a remainder. It’s helpful in algebraic manipulation, such as solving equations with fractions in them.
TLDR
- The Highest Common Factor (HCF) is the largest number that divides exactly into two or more numbers
- Main methods to find the HCF:
- Listing factors and finding the highest common one
- Using prime factorisation
- Using the division method (Euclidean algorithm)
- HCF is used for:
- Simplifying fractions (divide numerator and denominator by HCF)
- Finding the largest unit that can measure quantities exactly
- Working out arrangements where groups need to be of equal size
- Always check your answer by verifying it divides into all numbers with no remainder
What You Need to Know (Taken From The Specification)
- You need to understand the concepts and vocabulary of factors, common factors, and highest common factor
- This connects with prime factorisation and the unique factorisation theorem
- You should be able to apply these concepts to solve problems
This topic is in both Foundation and Higher tiers.
Key Facts
🔑 Key Fact: A factor of a number divides into that number exactly (with no remainder). So 2, 3, 5, 10 and 25 are factors of 50 – and 4, 6 and 9 aren’t.
🔑 Key Fact: The HCF of two numbers is the product of all their common prime factors, each raised to the minimum power they appear in either number.
🔑 Key Fact: A prime factor is a factor that is a prime number. So 2, 3 and 5 are prime factors of 50, with 10 and 25 being factors of 50 (but not prime factors).
🔑 Key Fact: The HCF of two numbers is always smaller than the smallest of the two original numbers.
Finding the HCF: Different Methods
There are several ways to find the HCF, each with its advantages. I’ll show you the main methods, and you can pick the one that works best for you.
Method 1: Listing Factors
This method works well for smaller numbers. Use it only when you quickly need to get the HCF. If a question specifically asks you to find the HCF, use method 2.
- List all factors of each number
- Identify the factors that appear in all lists
- The largest of these common factors is the HCF
Example 1
Find the HCF of 24 and 36.
Step 1: List all factors of each number.
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Step 2: Identify the common factors.
- Common factors: 1, 2, 3, 4, 6, 12
Step 3: Find the largest common factor.
- HCF = 12
Therefore, the HCF of 24 and 36 is 12.
Method 2: Prime factorisation
This method is very efficient, especially for larger numbers. It should be your default method, especially when a questions asks you to find the HCF – you will get method marks for using it.
- Find the prime factorisation of each number using a tree diagram (see below)
- Identify the prime factors common to all numbers
- Take each common prime factor to the minimum power it appears in any of the numbers
- Multiply these prime powers together to get the HCF
Example 2
Find the HCF of 36 and 48.
Step 1: Find the prime factorisation of each number.
- 36 = 2² × 3²
- 48 = 2⁴ × 3
Step 2: Identify the common prime factors and their minimum powers.
- Common prime factors: 2 and 3
- Minimum power of 2: min(2, 4) = 2
- Minimum power of 3: min(2, 1) = 1
Step 3: Multiply these prime powers to get the HCF.
- HCF = 2² × 3¹
- HCF = 4 × 3
- HCF = 12
Therefore, the HCF of 36 and 48 is 12.
Method 3: Division Method (Euclidean Algorithm)
This method is particularly useful for large numbers.
- Divide the larger number by the smaller number
- If the remainder is 0, the HCF is the smaller number
- If the remainder is not 0, divide the smaller number by the remainder
- Continue this process until the remainder is 0
- The last divisor is the HCF
Example 3
Find the HCF of 48 and 18.
Step 1: Divide the larger number by the smaller number.
- 48 ÷ 18 = 2 remainder 12
Step 2: Divide the previous divisor by the remainder.
- 18 ÷ 12 = 1 remainder 6
Step 3: Divide the previous divisor by the remainder.
- 12 ÷ 6 = 2 remainder 0
Step 4: Since the remainder is now 0, the HCF is the last divisor.
- HCF = 6
Therefore, the HCF of 48 and 18 is 6.
HCF of More Than Two Numbers
To find the HCF of more than two numbers, you can either:
- Find the HCF of the first two numbers, then find the HCF of that result and the third number, and so on.
- Use the prime factorisation method directly on all numbers.
Example 4
Find the HCF of 12, 18, and 30.
Method 1: Step-by-step HCF
- First, find HCF(12, 18) = 6
- Then, find HCF(6, 30) = 6
Method 2: Prime factorisation
- 12 = 2² × 3
- 18 = 2 × 3²
- 30 = 2 × 3 × 5
- Common prime factors: 2 and 3
- Minimum powers: 2¹ and 3¹
- HCF = 2 × 3 = 6
Therefore, the HCF of 12, 18, and 30 is 6.
Applications of HCF
Simplifying Fractions
To simplify a fraction, divide both the numerator and denominator by their HCF.
Example 5
Simplify the fraction 36/48.
Step 1: Find the HCF of 36 and 48.
- As we found in Example 2, HCF = 12
Step 2: Divide both numerator and denominator by the HCF.
- 36 ÷ 12 = 3
- 48 ÷ 12 = 4
Therefore, 36/48 = 3/4 in its simplest form.
Sharing Items Equally
The HCF helps determine the maximum number of equal groups that can be formed.
Example 6
A baker has 35 chocolate cupcakes and 49 vanilla cupcakes. She wants to make identical gift boxes, each containing the same number of chocolate cupcakes and the same number of vanilla cupcakes, with no cupcakes left over. What is the maximum number of gift boxes she can make?
Step 1: Find the HCF of 35 and 49.
- 35 = 5 × 7
- 49 = 7²
- Common prime factor: 7
- Minimum power: 7¹
- HCF = 7
Step 2: The maximum number of identical gift boxes is equal to the HCF.
- Maximum number of gift boxes = 7
Step 3: Calculate how many cupcakes of each type go in each box.
- Chocolate cupcakes per box = 35 ÷ 7 = 5
- Vanilla cupcakes per box = 49 ÷ 7 = 7
Therefore, the baker can make 7 identical gift boxes, each containing 5 chocolate cupcakes and 7 vanilla cupcakes.
Common Mistakes to Avoid
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Confusing HCF with LCM:
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Incorrect: Thinking that the HCF is the smallest number that is a multiple of both numbers
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Correct: The HCF is the largest number that divides exactly into both numbers
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Finding common multiples instead of common factors:
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Incorrect: Looking for common numbers in the lists of multiples
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Correct: Look for common numbers in the lists of factors
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Assuming that the HCF is always small:
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Incorrect: Thinking the HCF can’t be one of the original numbers
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Correct: The HCF can be equal to the smaller number if the smaller number divides the larger number exactly
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Making errors in prime factorisation:
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Incorrect: Missing prime factors or getting powers wrong
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Correct: Double-check your prime factorisations before finding common factors
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Forgetting to check the answer:
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Incorrect: Not verifying that your HCF divides both numbers exactly
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Correct: Always check your answer by dividing each original number by your HCF – there should be no remainder
Questions
Try these questions to practice finding the HCF:
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Find the HCF of 15 and 25.
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Calculate the HCF of 54 and 90.
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Find the HCF of 17 and 23.
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Determine the HCF of 144, 192, and 240.
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Simplify the fraction 56/72 by finding the HCF of the numerator and denominator.
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A florist has 48 roses, 60 tulips, and 84 lilies. She wants to make identical bouquets, each containing the same number of each type of flower, with no flowers left over. What is the maximum number of bouquets she can make?
Solutions
Question 1
Factors of 15: 1, 3, 5, 15
Factors of 25: 1, 5, 25
Common factors: 1, 5
HCF = 5
Alternative method using prime factorisation:
15 = 3 × 5
25 = 5²
Common prime factor: 5
Minimum power: 5¹
HCF = 5
Question 2
Using prime factorisation:
54 = 2 × 3³
90 = 2 × 3² × 5
Common prime factors: 2 and 3²
HCF = 2 × 3² = 2 × 9 = 18
Question 3
Prime factorisation:
17 is a prime number
23 is a prime number
They have no common prime factors except 1
HCF = 1
This means 17 and 23 are coprime.
Question 4
Using prime factorisation:
144 = 2⁴ × 3²
192 = 2⁶ × 3
240 = 2⁴ × 3 × 5
Common prime factors: 2⁴ and 3¹
HCF = 2⁴ × 3 = 16 × 3 = 48
Question 5
Find the HCF of 56 and 72:
56 = 2³ × 7
72 = 2³ × 3²
Common prime factor: 2³
HCF = 2³ = 8
Simplify the fraction:
56 ÷ 8 = 7
72 ÷ 8 = 9
Therefore, 56/72 = 7/9 in its simplest form.
Question 6
Find the HCF of 48, 60, and 84:
48 = 2⁴ × 3
60 = 2² × 3 × 5
84 = 2² × 3 × 7
Common prime factors: 2² and 3
HCF = 2² × 3 = 4 × 3 = 12
Therefore, the florist can make 12 identical bouquets, with:
- Roses per bouquet = 48 ÷ 12 = 4
- Tulips per bouquet = 60 ÷ 12 = 5
- Lilies per bouquet = 84 ÷ 12 = 7
Summary
- The Highest Common Factor (HCF) is the largest positive integer that divides exactly into two or more numbers
- There are several methods to find the HCF:
- Listing factors and identifying the largest common one
- Using prime factorisation and taking common primes to their minimum powers
- Using the division method (Euclidean algorithm)
- The HCF has many practical applications, including simplifying fractions and solving problems involving equal groups
- Remember that the HCF of two numbers is never larger than the smaller of the two numbers
- Two numbers with an HCF of 1 are called coprime
Finding the HCF might seem a bit mechanical at first, but with practice, you’ll be able to spot patterns and shortcuts. The prime factorisation method is particularly powerful once you get comfortable with it.
By Atomic