Every HCF/LCM question rests on one skill: prime factorisation — splitting a number into the primes that multiply to make it, like 360=23×32×5. Do that once and both the HCF and the LCM fall straight out.
HCF vs LCM — same factor tree, opposite rule
• HCF (Highest Common Factor): the biggest number that divides them all. Take only the primes common to every number, each at its lowest power.
• LCM (Least Common Multiple): the smallest number they all divide into. Take every prime that appears, each at its highest power.
Example with 12=22×3 and 18=2×32: HCF =21×31=6 (lowest powers), LCM =22×32=36 (highest powers).
• Big awkward numbers? Use the division (Euclidean) method for the HCF: divide, then keep dividing the previous divisor by the remainder — the last non-zero remainder is the HCF.
• Two quick checks: the HCF always divides the difference of any two numbers, and the LCM is never smaller than the largest number.
⚖️HCF & LCM MachineSame prime factors — lowest powers make the HCF, highest powers make the LCM
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Third (optional)
72=2³×3²
120=2³×3×5
HCF → only the primes shared by every number (highlighted), each at its LOWEST power.
LCM → every prime that appears anywhere, each at its HIGHEST power.
HCF=2³×3=24
LCM=2³×3²×5=360
Check: HCF × LCM = 24 × 360 = 8,640 a × b = 72 × 120 = 8,640 — they match ✓
This identity works for exactly two numbers — it fails for three or more.
Using lowest common powers, the HCF of 22×3 and 2×32 is 2×3= .
📝Practice Questions
Q1What is the LCM of 6, 8 and 12?
Q2What is the HCF of 18 and 24?
Q3The HCF of two co-prime numbers is always:
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Real exam questions — HCF & LCM Basics
4 question types · 15 solved examples from real SSC papers
HCF is the biggest factor shared by the numbers; LCM is the smallest multiple they all divide. Break into primes once and both fall out — lowest common powers for HCF, highest powers for LCM.
How to solve this type
Break each number into primes, then take the HIGHEST power of every prime that appears anywhere. Multiply those together — that is the LCM. Quick check: the LCM can never be smaller than the largest number, and it must be divisible by each number. For decimals, first turn them into fractions and use LCM=HCF of denominatorsLCM of numerators.
