Prime Factorisation & LCM/HCF Methods explained through a friendly whiteboard conversation, examples, visuals, and practice for school students.
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Notes
Step 2
Formulae
Step 3
Examples
Step 4
Practice
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HCF and LCM help us work with common factors and common multiples.
They are used in fractions, grouping, timing problems, and word problems.
Prime factorisation gives a clear method for finding both.
HCF means highest common factor. It is the greatest number that divides all the given numbers exactly.
LCM means lowest common multiple. It is the smallest positive number that is a multiple of all the given numbers.
For small numbers, listing factors or multiples can work. For larger numbers, prime factorisation is more reliable.
For two numbers, HCF x LCM equals the product of the two numbers.
Meaning: List factors of each number and choose the greatest common one.
Definition:
Factors of 12 are 1, 2, 3, 4, 6, 12. Factors of 18 are 1, 2, 3, 6, 9, 18. HCF is 6.
Meaning: List multiples and choose the first common one.
Definition: LCM is the smallest common multiple.
Multiples of 4 are 4, 8, 12, 16. Multiples of 6 are 6, 12, 18. LCM is 12.
Meaning: Break each number into prime factors.
Definition: Prime factorisation writes a number as a product of primes.
For HCF, take common primes with the smallest powers. For LCM, take all primes with the greatest powers.
Meaning: For two numbers, HCF and LCM are connected by multiplication.
Definition: HCF x LCroduct of the two numbers.
If you know three of these values, you can find the fourth.
Easy
Question: Find the HCF of 18 and 24.
Medium
Question: Find the LCM of 12 and 18 using prime factorisation.
Exam-level
Question: The HCF of two numbers is 6 and their LCM is 72. One number is 18. Find the other number.
Mixing HCF and LCM methods
Why it happens: Both use factors and multiples.
Correct approach: Remember: HCF is greatest shared factor. LCM is smallest shared multiple.
Using greatest powers for HCF
Why it happens: Students copy the LCM rule.
Correct approach: HCF uses common prime factors with smallest powers.
Using the product relation for more than two numbers
Why it happens: The formula is easy to remember.
Correct approach: Use it only for two numbers.
Can HCF be 1?
Yes. Numbers with HCF 1 are called crime.
Is LCM always bigger than both numbers?
Not always. If one number is a multiple of the other, the LCM is the bigger number.
Which method is best?
Listing is fine for small numbers. Prime factorisation is better for larger numbers.
Practise the same concept from this article. Your student profile may guide entitlement and follop recommendations, but it does not replace this concept.