Ever wondered how to find what numbers have in common... Show more
Maths81 views·Updated May 30, 2026·3 pagesUnderstanding HCF and LCM: Step-by-Step Guide
Jess Davies@jess_notes21
1 / 3
1
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Understanding Prime Factors and Factor Trees
Prime numbers are the building blocks of all numbers - they only divide by themselves and 1 (like 2, 3, 5, 7, 11). Think of them as mathematical atoms that can't be broken down further.
A prime factor tree helps you break any number down into these basic building blocks. For example, to find the prime factors of 72, you keep dividing until you're left with only prime numbers.
Starting with 72, you might divide by 8 and 9, then break those down further until you get 72 = 3 × 3 × 2 × 2 × 2. In index form, this becomes 3² × 2³, which is much tidier to write.
Quick Tip: Always start your factor tree with any factors you can spot easily - there's no single "correct" way to draw it, as long as you end up with prime numbers at the bottom!
2
of 3
Finding HCF and LCM Using Prime Factors
The highest common factor (HCF) is the biggest number that divides into both your numbers exactly. The lowest common multiple (LCM) is the smallest number that both your original numbers divide into.
Let's work through finding the HCF and LCM of 36 and 56. First, break both numbers into their prime factors using factor trees: 36 = 3 × 3 × 2 × 2 and 56 = 7 × 2 × 2 × 2.
Now comes the clever bit - draw a Venn diagram with two overlapping circles. Put the prime factors that appear in both numbers (the common ones) in the overlapping section.
Remember: The Venn diagram method makes it impossible to get HCF and LCM mixed up - you'll never confuse which calculation to use!
3
of 3
Calculating Your Final Answers
For 36 and 56, the common prime factors are 2 × 2, so these go in the overlapping section of your Venn diagram. The remaining factors (3 × 3 for 36, and 7 × 2 for 56) go in the outer parts of each circle.
The HCF is simply the product of everything in the overlap: 2 × 2 = 4. This means 4 is the largest number that divides exactly into both 36 and 56.
The LCM uses every prime factor exactly once, so multiply everything across both circles: 3 × 3 × 2 × 2 × 2 × 7 = 504. This means 504 is the smallest number that both 36 and 56 divide into exactly.
Pro Tip: You can check your LCM by verifying that both original numbers divide into it exactly - 504 ÷ 36 = 14 and 504 ÷ 56 = 9, both with no remainders!
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Maths81 views·Updated May 30, 2026·3 pagesUnderstanding HCF and LCM: Step-by-Step Guide
Jess Davies@jess_notes21
Ever wondered how to find what numbers have in common or their smallest shared multiple? Understanding HCF and LCM is crucial for solving fraction problems, simplifying expressions, and tackling exam questions with confidence.
1
of 3Sign up to see the content. It's free!
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Understanding Prime Factors and Factor Trees
Prime numbers are the building blocks of all numbers - they only divide by themselves and 1 (like 2, 3, 5, 7, 11). Think of them as mathematical atoms that can't be broken down further.
A prime factor tree helps you break any number down into these basic building blocks. For example, to find the prime factors of 72, you keep dividing until you're left with only prime numbers.
Starting with 72, you might divide by 8 and 9, then break those down further until you get 72 = 3 × 3 × 2 × 2 × 2. In index form, this becomes 3² × 2³, which is much tidier to write.
Quick Tip: Always start your factor tree with any factors you can spot easily - there's no single "correct" way to draw it, as long as you end up with prime numbers at the bottom!
2
of 3Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
Finding HCF and LCM Using Prime Factors
The highest common factor (HCF) is the biggest number that divides into both your numbers exactly. The lowest common multiple (LCM) is the smallest number that both your original numbers divide into.
Let's work through finding the HCF and LCM of 36 and 56. First, break both numbers into their prime factors using factor trees: 36 = 3 × 3 × 2 × 2 and 56 = 7 × 2 × 2 × 2.
Now comes the clever bit - draw a Venn diagram with two overlapping circles. Put the prime factors that appear in both numbers (the common ones) in the overlapping section.
Remember: The Venn diagram method makes it impossible to get HCF and LCM mixed up - you'll never confuse which calculation to use!
3
of 3Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
Calculating Your Final Answers
For 36 and 56, the common prime factors are 2 × 2, so these go in the overlapping section of your Venn diagram. The remaining factors (3 × 3 for 36, and 7 × 2 for 56) go in the outer parts of each circle.
The HCF is simply the product of everything in the overlap: 2 × 2 = 4. This means 4 is the largest number that divides exactly into both 36 and 56.
The LCM uses every prime factor exactly once, so multiply everything across both circles: 3 × 3 × 2 × 2 × 2 × 7 = 504. This means 504 is the smallest number that both 36 and 56 divide into exactly.
Pro Tip: You can check your LCM by verifying that both original numbers divide into it exactly - 504 ÷ 36 = 14 and 504 ÷ 56 = 9, both with no remainders!
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