LCM, HCF and Their Applications
LCM, HCF and Their Applications
LCM and HCF give 2 to 3 questions in the Part-III maths paper, and unlike the Part-II version these go beyond simple bell problems into remainders and fraction forms. The method never changes; only the wrapper does. Learn to read what the situation is asking for.
Finding HCF and LCM
Two methods matter. Prime factorisation suits questions asking for reasoning; the division method is faster in the hall.
For 12 and 18: 12 = 2² × 3 and 18 = 2 × 3².
- HCF takes the lowest power of every common prime: 2 × 3 = 6.
- LCM takes the highest power of every prime present: 2² × 3² = 36.
HCF × LCM = Product of the two numbers
This identity holds for two numbers only. If the HCF of two numbers is 6, their LCM is 36 and one number is 12, the other must be 18. For three or more numbers it fails, and that failure is itself asked.
Choosing Between Them
| The question says | You need |
|---|---|
| Largest number that divides ... | HCF |
| Greatest length or maximum equal pieces | HCF |
| Largest container measuring several quantities | HCF |
| Least number divisible by ... | LCM |
| Events happening together again | LCM |
| Smallest number leaving the same remainder | LCM |
✗ Cutting three ropes into the longest equal pieces needs the LCM | ✓ It needs the HCF — the answer must divide all three lengths
The Remainder Cases
Three patterns cover almost every hard question in this topic.
- Greatest number dividing a, b, c and leaving remainders r1, r2, r3 → take the HCF of (a − r1), (b − r2), (c − r3).
- Greatest number dividing a, b, c and leaving the same remainder → take the HCF of the differences (b − a) and (c − b).
- Least number which when divided by a, b, c leaves remainder r each time → LCM of a, b, c, then add r.
Worked example: the least number that leaves remainder 3 when divided by 6, 8 and 12. LCM of 6, 8, 12 is 24, so the answer is 27.
LCM and HCF of Fractions
HCF of fractions = HCF of numerators ÷ LCM of denominators
LCM of fractions = LCM of numerators ÷ HCF of denominators
Note the crossing: numerators and denominators take opposite operations. For 2/3 and 4/9, the HCF is HCF(2,4) ÷ LCM(3,9) = 2/9. Candidates who apply the same operation to both parts get the standard wrong option.
सबसे बड़ा जो बांटे = म.स.प.; सबसे छोटा जो विभाज्य हो = ल.स.प.।
TRE pointer: In Part-III the plain bell question is rare; the remainder variants and the fraction formulas are what actually appear, because they separate a candidate who memorised a method from one who understands it. The crossed formula for fractions is the single most missed item in this topic. With five options and −1/3 negative marking, work the remainder subtraction on paper rather than trusting a quick reading of the question.
60-Second Recap
- HCF takes lowest powers of common primes; LCM takes highest powers of all primes.
- HCF × LCM = product of the numbers — two numbers only.
- Largest or maximum wording means HCF; together again or least means LCM.
- With different remainders, subtract each remainder first, then take the HCF.
- With the same remainder unknown, take the HCF of the differences.
- HCF of fractions = HCF of numerators over LCM of denominators; LCM crosses the other way.
Frequently Asked Questions
How do I find the HCF of fractions?
Take the HCF of the numerators and divide it by the LCM of the denominators. The LCM of fractions works the other way round: LCM of numerators over HCF of denominators. Applying the same operation to both parts is the error the options are designed to catch.
What is the least number that leaves the same remainder with several divisors?
Find the LCM of the divisors and add the required remainder. For a remainder of three with divisors six, eight and twelve, the LCM is twenty-four, so the answer is twenty-seven. Verify by dividing: twenty-seven leaves three in all three cases.
Does HCF times LCM equal the product for three numbers?
No. The identity works only for two numbers. For three or more, use prime factorisation instead, taking lowest powers for the HCF and highest powers for the LCM. Questions sometimes supply three numbers precisely to see whether you apply the shortcut wrongly.
How do I decide whether a word problem needs LCM or HCF?
Look at the shape of the answer. If it must be larger than the given numbers, as with events recurring together, it is the LCM. If it must divide into them, as with cutting the longest equal pieces or filling the largest container, it is the HCF.