LCM and HCF made clear.
HCF finds the largest equal grouping size. LCM finds the first common meeting point of repeating quantities. Recognising which situation you have is more useful than memorising a keyword.
Key concepts in LCM and HCF
HCF or GCD
The greatest positive integer dividing all the given positive integers exactly.
LCM
The smallest positive integer divisible by all the given positive integers.
Prime powers
For HCF use common primes with their lowest powers. For LCM use every prime with its highest power.
Two-number identity
For two positive integers a,b: HCF(a,b) × LCM(a,b) = ab. This product identity does not extend unchanged to three numbers.
Prime-power comparison
| Prime | 18 | 24 | HCF uses | LCM uses |
|---|---|---|---|---|
| 2 | 2¹ | 2³ | 2¹ | 2³ |
| 3 | 3² | 3¹ | 3¹ | 3² |
| Result | 18 | 24 | 6 | 72 |
A reliable solving method
- Decide whether you are dividing into equal groups or finding a shared multiple.
- Prime-factorise the numbers or use repeated division.
- Choose minimum powers for HCF and maximum powers for LCM.
- Check that the HCF divides each input and the LCM is divisible by each.
Step-by-step solved examples
Find HCF and LCM of 18 and 24.
- 18=2×3²; 24=2³×3.
- HCF=2×3=6.
- LCM=2³×3²=72; check 6×72=18×24.
Bells ring every 12 and 18 minutes, together at noon. Next together?
- A common ringing time must be a multiple of both intervals.
- LCM(12,18)=36 minutes.
- The next simultaneous ring is at 12:36 pm.
Cut 48 cm and 60 cm ribbons into the longest equal pieces without waste.
- The piece length must divide both 48 and 60.
- HCF(48,60)=12 cm.
- This gives 4+5=9 equal pieces.
LCM and HCF: practice questions
Choose one answer before opening the explanation. Write down your reasoning as well as the option; a correct guess is different from a method you can repeat.
HCF of 12 and 20?
Show answer and explanation — question 1
Answer B: 4
Common factors are 1,2,4; the greatest is 4.
LCM of 6 and 8?
Show answer and explanation — question 2
Answer C: 24
24 is the smallest positive multiple of both.
HCF of two coprime positive integers?
Show answer and explanation — question 3
Answer B: 1
Coprime means their greatest common divisor is 1.
HCF 5, LCM 60, one number 15. Other?
Show answer and explanation — question 4
Answer C: 20
Other = 5×60/15 = 20.
Three lights flash every 4,6,10 seconds. Together again after?
Show answer and explanation — question 5
Answer D: 60 s
LCM = 2²×3×5 = 60 seconds.
Largest equal square tile for a 24 cm by 36 cm rectangle?
Show answer and explanation — question 6
Answer C: 12 cm
HCF(24,36)=12 cm, assuming no cuts or gaps.
Frequently asked questions
Can the HCF exceed the smaller positive number?
No. A divisor cannot exceed the positive integer it divides.
What if one number divides another?
The smaller is the HCF and the larger is the LCM.
Decide whether you are dividing into equal groups or finding a shared multiple. Check that the HCF divides each input and the LCM is divisible by each.