HCF and LCM
- CBSE · NCERT Class 10, Ch 1 Real Numbers
The HCF of two numbers is the largest number that divides both exactly, and the LCM is the smallest number that both divide into exactly.
HCF stands for Highest Common Factor. LCM stands for Lowest Common Multiple. Class 10 finds both through prime factorisation. For the HCF, take each prime the numbers share, at its smallest power. For the LCM, take every prime that appears, at its largest power. Two numbers whose HCF is 1 are called co-prime numbers.
A small example
Take 12 and 18. In primes, 12 = 2² × 3 and 18 = 2 × 3².
- HCF = 2 × 3 = 6, the smaller power of 2 and of 3.
- LCM = 2² × 3² = 36, the larger power of each.
Check: 6 × 36 = 216, and 12 × 18 = 216. For any two positive whole numbers, HCF × LCM equals their product. The chapter points out that this does not hold for three numbers.
Where marks go
Word problems do not say HCF or LCM. You have to spot which one is asked. Question 21(A) of CBSE's 2026-27 Maths Standard sample paper asks for the longest ruler that measures a hall's length, breadth and height exactly. The hall is 975 cm by 675 cm by 525 cm.
The board's Maths Standard marking scheme for 2026-27 gives half a mark for seeing this as an HCF question. It gives 1 mark for the three prime factorisations. The last half mark is for the answer, 75 cm. The internal choice in that question, three bells that toll together, asks for an LCM instead.
Sources
Facts last checked against these sources on 30 September 2026.
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