LCM and HCF are two of the most reliably tested topics in SSC MTS Quant. If you get these right every time, you are picking up almost free marks — the question types repeat with minor numerical changes year after year.
Here is the plain-language version:
HCF (Highest Common Factor) — also called GCD (Greatest Common Divisor) — is the largest number that divides two or more numbers without leaving a remainder. Think of it as the biggest piece you can cut all your numbers into evenly.
LCM (Least Common Multiple) is the smallest number that all your given numbers divide into evenly. Think of it as the smallest meeting point — the first place where all numbers "sync up".
A useful analogy: imagine two runners on a circular track. Runner A takes 4 minutes per lap, Runner B takes 6 minutes. They both start at the same point. When do they meet again at the start? At the LCM of 4 and 6, which is 12 minutes. That is the "sync point" interpretation of LCM.
Now flip it. You have two ropes of length 40 cm and 60 cm. You want to cut both into pieces of equal length without any leftover. What is the longest possible piece? That is the HCF of 40 and 60, which is 20 cm. HCF is the "biggest common piece" idea.
One relationship you must know cold:
This only works for exactly two numbers, not three or more. Use it freely for two-number problems.
SSC MTS loves three question types: direct LCM/HCF of integers, LCM/HCF of fractions, and the "largest number that divides with given remainders" trap. All three appear in recent papers. The concept section above covers the mental model — the Deep Dive below covers the exact mechanics of each.
This is your primary tool. Factor each number into prime factors, then apply the rule:
Example: Find HCF and LCM of 36, 48, and 60.
When the numbers are large, divide the bigger by the smaller, then divide the previous divisor by the remainder. Repeat until remainder is 0. The last non-zero divisor is the HCF.
Example: HCF(220, 2024)
HCF = 44. (This is Euclid's algorithm and it is faster than factoring when numbers are large like in SSC questions involving 2000+ range numbers.)
SSC MTS has tested this multiple times. The rule is fixed — memorize it:
This is counterintuitive — HCF divides by LCM and LCM divides by HCF. Many candidates flip this and lose the mark.
Two sub-types appear in SSC exams:
Sub-type A — Same remainder: If a number N divides 38 and 62 both leaving remainder 2, find N. Answer: HCF(38 − 2, 62 − 2) = HCF(36, 60) = 12.
Sub-type B — Different remainders: If N divides 2036 leaving remainder 12, and divides 233 leaving remainder 13, find the largest such N. Subtract: (2036 − 12) = 2024, (233 − 13) = 220. Answer: HCF(2024, 220).
The logic: after removing the remainder, N must divide the cleaned number exactly. So N divides both cleaned numbers, making it a common factor. Largest common factor = HCF.
If two numbers are in ratio a : b and their HCF is h, then the numbers are ah and bh. Their LCM = abh (since LCM × HCF = product of two numbers, and product = ah × bh = abh², so LCM = abh). Their sum = h(a + b).
This is a very efficient framework when ratio and HCF are given together — which SSC MTS has tested as recently as 2025.
Any two distinct prime numbers share no common factor other than 1 — by definition of primality. So HCF of any two distinct primes is always 1. This is a direct-mark question when it appears.
For fractions, LCM and HCF behave opposite to what you expect. LCM of fractions uses HCF of denominators (not LCM). HCF of fractions uses LCM of denominators (not HCF). Anchor this with: "fractions flip the bottom." Test: LCM of 2/3 and 5/6. LCM(2,5)=10, HCF(3,6)=3. Answer = 10/3. Standard approach of trying to remember by logic mid-exam: 45 seconds. Using this anchor: 10 seconds.
When ratio a:b and HCF = h are given, the numbers are ah and bh. Skip all factorization. Example: ratio 5:9, HCF = 7 → numbers are 35 and 63 immediately. Sum = 7(5+9) = 7×14 = 98. Standard method (find numbers by trial, verify HCF): 6 steps. This substitution: 2 steps.
For two large numbers, use successive division instead of factorization. Divide bigger by smaller, replace bigger with remainder, repeat. The final non-zero remainder is HCF. For HCF(2024, 220): 2024 = 9×220 + 44, then 220 = 5×44 + 0. HCF = 44. Prime factorization of 2024 would take 8+ steps; Euclid takes 2 steps here.
"Largest number dividing A and B leaving remainders r₁ and r₂" — always subtract the remainder from each number first, then find HCF of the cleaned values. No formula to memorize — just the logic: after cleaning, the number must divide exactly, so it is a common factor. Sub-type B (different remainders): HCF(A − r₁, B − r₂). Sub-type A (same remainder r): HCF(A − r, B − r) or equivalently HCF(B − A). This saves the 20 seconds candidates waste setting up equations.
When one number is a multiple of another, LCM = the larger number. LCM(15, 75) = 75. LCM(12, 36) = 36. No calculation needed — check divisibility first before factorizing. This single check saves 30 seconds on about one in four LCM questions.
Read the question stem for these triggers:
"Greatest number that divides / largest number that exactly divides" → HCF problem. If remainders are given, subtract them first. Use Euclid's algorithm if numbers exceed 200.
"Smallest / minimum distance / least number / when do they meet" → LCM problem.
"Ratio given, HCF given" → Multiply ratio terms by HCF to get the numbers. Done in one step.
"Fractions in the question" → Apply the fraction rule (LCM of numerators over HCF of denominators, or HCF of numerators over LCM of denominators). Do not forget the "flip the bottom" pattern.
"Two prime numbers" → HCF = 1 automatically, LCM = their product.
"Verify your answer" → LCM × HCF = Product of two numbers. Use this as a 5-second check when you have time.
For three or more numbers, use prime factorization — Euclid and the product shortcut only work for pairs.
Why this question: LCM of fractions is a guaranteed SSC MTS repeat — it tests whether you have the fraction rule locked in or not.
Solving path: Apply the fraction LCM rule. Numerators: 2, 5, 5, 7. LCM(2, 5, 5, 7) — no shared primes except the repeated 5, so LCM = 2 × 5 × 7 = 70. Denominators: 3, 6, 12, 18. HCF(3, 6, 12, 18) — the smallest is 3 and it divides all, so HCF = 3. Answer = 70/3.
Why this question: "Greatest number that exactly divides" is the standard HCF word problem. Spot it, factor all three, find HCF.
Solving path: Factor each: 38 = 2 × 19, 95 = 5 × 19, 171 = 9 × 19. The common factor in all three is 19. HCF = 19. You can also notice 95 − 38 = 57 = 3 × 19, and 171 − 95 = 76 = 4 × 19, confirming 19.
Why this question: Three-number LCM via prime factorization — the most mechanical form of LCM question. Knowing to take highest powers is the only skill tested.
Solving path: 16 = , 20 = , 24 = . LCM = .
Why this question: Step-length / distance problems are an SSC MTS favorite. The "minimum distance in complete steps" phrasing always means LCM.
Solving path: LCM(77, 88). Factor: 77 = 7 × 11, 88 = . LCM = cm.
Why this question: This 2020 question combines remainder subtraction with HCF via Euclid — it is the hardest variant in this topic cluster and worth understanding thoroughly.
Solving path: Clean the numbers: 2036 − 12 = 2024, 233 − 13 = 220. Now find HCF(2024, 220) using Euclid: 2024 = 9 × 220 + 44; 220 = 5 × 44 + 0. HCF = 44. Verify: 2024 ÷ 44 = 46 (exact), 220 ÷ 44 = 5 (exact). Answer = 44.
Flipping the fraction rule. Writing LCM of fractions as "LCM of numerators / LCM of denominators" — this is wrong. The denominator uses HCF. This mistake alone causes wrong answers on a question that should take 20 seconds.
Using the product shortcut on three numbers. LCM × HCF = Product of numbers is valid only for two numbers. Applying it to three numbers gives an incorrect result every time.
Forgetting to subtract the remainder before finding HCF. In "largest number dividing A leaving remainder r" problems, many candidates find HCF(A, B) directly. You must find HCF(A − r₁, B − r₂) first.
Treating HCF of two primes as 2. This is a trap option in MCQs. The HCF of any two distinct primes is 1, not 2. The option "2" is there to catch you if you confuse "both are primes" with "both are even primes."
Confusing ratio terms with the actual numbers. If ratio is 5:9 and HCF is 7, the numbers are 35 and 63 — not 5 and 9. Many candidates compute HCF(5, 9) and get 1, then wonder why it does not match.
Not checking whether one number divides another before factorizing. If LCM(24, 72) is asked, 72 = 3 × 24, so LCM = 72 directly. Missing this check costs 30 seconds on what should be a 5-second problem.