LCM and HCF Tricks for SSC CGL: The Short Answer
HCF (Highest Common Factor) is the largest number that divides two or more numbers exactly, and LCM (Least Common Multiple) is the smallest number that all of them divide exactly. The fastest exam method is prime factorisation: HCF = product of common primes with the lowest powers, LCM = product of all primes with the highest powers. For two numbers, always remember HCF × LCM = product of the numbers.
Key takeaway for SSC: SSC CGL, CHSL, MTS and GD ask 1 to 2 LCM and HCF questions in the maths section, almost always in one of six patterns: product formula, fractions, "same remainder", "different remainders", bells or lights together, and largest tile or container. Learn the six rules in the revision table below and you can answer each one in under 40 seconds. Every answer on this page was checked with a Python program.
LCM and HCF sit inside the number system chapter, so pair this note with our unit digit tricks for SSC CGL and surds and indices for SSC CGL.
What Is LCM and HCF? (Meaning with Example)
Take 12 and 18.
- Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. Common factors are 1, 2, 3, 6, so HCF = 6.
- Multiples of 12: 12, 24, 36, 48 … Multiples of 18: 18, 36, 54 … The first common multiple is 36, so LCM = 36.
| Term | Full form | Other names | Always … |
|---|---|---|---|
| HCF | Highest Common Factor | GCD (Greatest Common Divisor), GCF | ≤ the smallest number |
| LCM | Least Common Multiple | Lowest Common Multiple | ≥ the largest number |
Two quick facts SSC likes to test:
- HCF always divides the LCM. If an option has an "HCF" that does not divide the "LCM", that pair is impossible.
- Co-prime numbers (HCF = 1), such as 8 and 15, have LCM = their product (120).
HCF and LCM Formula by Prime Factorisation Method
Write each number as a product of prime powers.
- HCF = product of the common primes, each with the lowest power.
- LCM = product of all primes that appear, each with the highest power.
Example 1: Find the HCF and LCM of 72 and 108.
72 = 2³ × 3² and 108 = 2² × 3³.
- HCF = 2² × 3² = 4 × 9 = 36
- LCM = 2³ × 3³ = 8 × 27 = 216
- Check: 36 × 216 = 7,776 = 72 × 108. ✔
Example 2 (three numbers): Find the HCF and LCM of 24, 36 and 60.
24 = 2³ × 3, 36 = 2² × 3², 60 = 2² × 3 × 5.
- HCF = 2² × 3 = 12
- LCM = 2³ × 3² × 5 = 360
Note that the product rule (HCF × LCM = product) works only for two numbers. Here 12 × 360 = 4,320, but 24 × 36 × 60 = 51,840.
HCF by Division Method (Euclid's Trick for Big Numbers)
When numbers are large, factorising is slow. Use repeated division: divide the larger number by the smaller, then divide the divisor by the remainder, and keep going until the remainder is 0. The last divisor is the HCF.
Example 3: HCF of 1,365 and 1,560.
| Step | Division | Remainder |
|---|---|---|
| 1 | 1,560 ÷ 1,365 = 1 | 195 |
| 2 | 1,365 ÷ 195 = 7 | 0 |
HCF = 195. Two steps instead of factorising both numbers.
For three numbers, find the HCF of the first two, then the HCF of that result with the third.
LCM and HCF Formula: Product of Two Numbers
For any two numbers a and b:
a × b = HCF(a, b) × LCM(a, b)
Example 4: The HCF of two numbers is 12 and their LCM is 360. If one number is 72, find the other.
Other number = (12 × 360) ÷ 72 = 60.
Example 5 (ratio type): Two numbers are in the ratio 3 : 4 and their LCM is 180. Find the numbers.
Let the numbers be 3x and 4x. Since 3 and 4 are co-prime, HCF = x and LCM = 12x. So 12x = 180, x = 15, and the numbers are 45 and 60.
Rule: if two numbers are in the ratio a : b (a and b co-prime), then HCF = x and LCM = a × b × x.
HCF and LCM of Fractions Formula
| Find | Formula |
|---|---|
| HCF of fractions | HCF of numerators ÷ LCM of denominators |
| LCM of fractions | LCM of numerators ÷ HCF of denominators |
Write each fraction in its lowest terms first.
Example 6: HCF and LCM of 2/3, 4/9 and 8/15.
- HCF = HCF(2, 4, 8) ÷ LCM(3, 9, 15) = 2 ÷ 45 = 2/45
- LCM = LCM(2, 4, 8) ÷ HCF(3, 9, 15) = 8 ÷ 3 = 8/3
For decimals, make the decimal places equal (0.6, 0.90, 1.2 → 60, 90, 120 hundredths), find the HCF or LCM of the whole numbers, then put the decimal back.
LCM and HCF Remainder Questions for SSC CGL (Most Asked)
These four rules cover nearly every remainder question.
| Question type | Use | Formula |
|---|---|---|
| Least number that leaves the same remainder r when divided by a, b, c | LCM | LCM(a, b, c) + r |
| Least number that leaves remainders where (divisor − remainder) is the same value k | LCM | LCM(a, b, c) − k |
| Greatest number that divides x, y, z leaving the same remainder (remainder unknown) | HCF | HCF of the differences (y − x), (z − y), (z − x) |
| Greatest number that divides x and y leaving remainders r₁ and r₂ | HCF | HCF(x − r₁, y − r₂) |
Example 7: Find the least number which when divided by 12, 15 and 20 leaves remainder 5 in each case.
LCM(12, 15, 20) = 60, so the number = 60 + 5 = 65.
Example 8: Find the least number which when divided by 6, 9 and 12 leaves remainders 4, 7 and 10 respectively.
Each remainder is 2 less than its divisor (6 − 4 = 9 − 7 = 12 − 10 = 2). So the number = LCM(6, 9, 12) − 2 = 36 − 2 = 34. Check: 34 ÷ 6 leaves 4, 34 ÷ 9 leaves 7, 34 ÷ 12 leaves 10. ✔
Example 9: Find the greatest number which divides 43, 91 and 183 leaving the same remainder.
Differences: 91 − 43 = 48, 183 − 91 = 92, 183 − 43 = 140. HCF(48, 92, 140) = 4.
Example 10: Find the greatest number that divides 1,657 and 2,037 leaving remainders 6 and 5 respectively.
HCF(1,657 − 6, 2,037 − 5) = HCF(1,651, 2,032) = 127.
Example 11 (combined condition): Find the least number which leaves remainder 3 when divided by 5, 6, 7 or 8, but is exactly divisible by 9.
LCM(5, 6, 7, 8) = 840, so the number is 840k + 3. Try k = 1, 2, 3 … and test divisibility by 9 (digit sum):
- k = 1: 843 → digit sum 15, no
- k = 2: 1,683 → digit sum 18, yes
Answer: 1,683.
LCM Questions on Bells, Lights and Runners
When events repeat at fixed intervals, they happen together again after the LCM of the intervals.
Example 12: Three bells toll at intervals of 6, 8 and 12 seconds. They toll together at 9:00 am. When will they next toll together, and how many times will they toll together in 30 minutes (counting the start)?
- LCM(6, 8, 12) = 24 seconds, so next together at 9:00:24 am.
- 30 minutes = 1,800 seconds. 1,800 ÷ 24 = 75 more times, plus the start = 76 times.
Watch the wording: "including the first toll" adds 1; "after the start" does not.
HCF Questions on Tiles, Rods and Containers
When something must be cut, filled or tiled with the largest equal size, use the HCF.
Example 13: A floor is 1,517 cm long and 902 cm wide. Find the largest square tile that fits exactly, and the number of tiles needed.
- HCF(1,517, 902) = 41 cm (1,517 = 37 × 41, 902 = 22 × 41).
- Tiles = (1,517 ÷ 41) × (902 ÷ 41) = 37 × 22 = 814 tiles.
Example 14 (smallest or largest number of given digits): Find the smallest 4-digit number exactly divisible by 12, 15 and 18.
LCM = 180. The first multiple of 180 above 1,000 is 180 × 6 = 1,080. The largest 4-digit one is 180 × 55 = 9,900.
LCM and HCF Short Tricks (Quick Revision Table)
| Clue in the question | Use | Quick formula |
|---|---|---|
| "greatest", "largest", "maximum size" | HCF | — |
| "least", "smallest", "together again", "minimum" | LCM | — |
| HCF and LCM given, one number given | Product rule | Other = (HCF × LCM) ÷ given |
| Numbers in ratio a : b with LCM L | Ratio rule | x = L ÷ (a × b) |
| Same remainder r, find least number | LCM | LCM + r |
| (Divisor − remainder) constant k | LCM | LCM − k |
| Same remainder, find greatest divisor | HCF | HCF of differences |
| Remainders r₁, r₂ given, find greatest divisor | HCF | HCF(x − r₁, y − r₂) |
| Fractions | Both | HCF = HCF(num) ÷ LCM(den); LCM = LCM(num) ÷ HCF(den) |
Common Mistakes in LCM and HCF Questions
- Using the product rule for three numbers. HCF × LCM = product holds only for two numbers.
- Forgetting to reduce fractions before applying the fraction formula (6/8 must become 3/4).
- Mixing up "greatest" and "least". Greatest divisor → HCF; least number → LCM.
- Counting bells wrongly. Decide whether the starting toll is included before adding 1.
- Not checking options. If the HCF given in an option does not divide the LCM, reject it at once.
LCM and HCF Questions for SSC CGL (Practice Set with Answers)
- Find the HCF and LCM of 84 and 126. Answer: HCF 42, LCM 252. 84 = 2² × 3 × 7, 126 = 2 × 3² × 7.
- Find the HCF of 391 and 667. Answer: 23. 667 ÷ 391 leaves 276; 391 ÷ 276 leaves 115; 276 ÷ 115 leaves 46; 115 ÷ 46 leaves 23; 46 ÷ 23 leaves 0.
- The HCF of two numbers is 15 and their LCM is 2,100. If one number is 105, find the other. Answer: 300. (15 × 2,100) ÷ 105 = 300.
- Find the least number which when divided by 16, 20 and 24 leaves remainder 7 in each case. Answer: 247. LCM = 240, so 240 + 7.
- Find the greatest number which divides 1,305, 4,665 and 6,905 leaving the same remainder in each case. Answer: 1,120. HCF(3,360, 2,240, 5,600) = 1,120; the common remainder is 185.
- Four bells ring at intervals of 4, 6, 10 and 15 seconds. After ringing together, how many more times will they ring together in one hour? Answer: 60 times. LCM = 60 s; 3,600 ÷ 60 = 60.
- Find the HCF and LCM of 3/5, 9/10 and 6/25. Answer: HCF 3/50, LCM 18/5. HCF(3, 9, 6) = 3, LCM(5, 10, 25) = 50; LCM(3, 9, 6) = 18, HCF(5, 10, 25) = 5.
- Find the least number exactly divisible by 18, 24 and 32. Answer: 288. 2⁵ × 3² = 288.
- Find the largest 5-digit number exactly divisible by 10, 12 and 15. Answer: 99,960. LCM = 60; 99,999 ÷ 60 leaves 39, so 99,999 − 39.
- Find the HCF of 2³ × 3² × 5 and 2² × 3³ × 7. Answer: 36. Common primes with lowest powers: 2² × 3² = 36.
For timed practice, take the SSC CGL Number System Mock Test 01 and the SSC CGL Maths Mini Mock Test 04.
LCM and HCF FAQs
What is the formula of LCM and HCF?
For two numbers, LCM × HCF = product of the two numbers. By prime factorisation, HCF uses the common primes with the lowest powers and LCM uses all primes with the highest powers.
How do you find the HCF and LCM of three numbers?
Factorise all three into primes. HCF is the product of primes common to all three with the lowest powers; LCM is the product of every prime with the highest power. Do not use the product formula for three numbers.
What is the HCF and LCM formula for fractions?
HCF of fractions = HCF of numerators ÷ LCM of denominators. LCM of fractions = LCM of numerators ÷ HCF of denominators. Reduce each fraction to lowest terms first.
How do I know whether to use LCM or HCF in a word problem?
Words like greatest, largest or maximum size point to HCF. Words like least, smallest, minimum or together again point to LCM.
How many LCM and HCF questions come in SSC CGL?
Usually 1 to 2 questions in SSC CGL Tier 1 maths, often merged with remainders or divisibility from the number system chapter. SSC CHSL, MTS and GD follow the same pattern.
Can HCF be greater than LCM?
No. HCF is always less than or equal to LCM, and HCF always divides LCM exactly. They are equal only when all the numbers are the same.
Practice Next
- Mock tests: SSC CGL Number System Mock Test 02, SSC CGL Number System Mock Test 03, SSC CGL Number System Mock Test 04, SSC GD Number System Mock Test 01 and SSC CHSL Maths Mini Mock Test 04.
- Related notes: unit digit tricks for SSC CGL, surds and indices for SSC CGL, ratio and proportion for SSC CGL and simplification tricks for bank exams.
- Exam details: SSC CGL 2026 and SSC CHSL 2026.
- Browse all study notes.