LCM and HCF Questions for SSC MTS

What is HCF (Highest Common Factor)?

HCF is the largest number that divides two or more numbers without leaving a remainder. Think of it as the biggest common 'chunk' you can break all numbers into. For example, if you have 12 apples and 18 oranges and want to pack them into identical boxes with no leftover, the largest box size is HCF(12, 18) = 6. HCF is also called GCD (Greatest Common Divisor).

• HCF is always less than or equal to the smallest number
• HCF of two coprime numbers (no common factor except 1) is always 1
• Find HCF by prime factorization: take the lowest power of each common prime
• Euclidean algorithm is faster for large numbers: HCF(a, b) = HCF(b, a mod b)

What is LCM (Least Common Multiple)?

LCM is the smallest number that is divisible by two or more given numbers. Think of it as the smallest common 'meeting point' for all numbers. For example, if bells ring every 12 and 18 seconds, they ring together every LCM(12, 18) = 36 seconds. LCM is used when you need a common time, distance, or quantity.

• LCM is always greater than or equal to the largest number
• LCM of coprime numbers is their product
• Find LCM by prime factorization: take the highest power of each prime that appears
• LCM is useful in problems about cycles, repetition, and 'meeting together'

The Golden Relationship

The most important formula in this topic is: LCM × HCF = Product of the two numbers. This is a lifesaver in SSC MTS because if you know any three values, you can instantly find the fourth. This relationship works ONLY for two numbers, not three or more.

• Always verify: LCM(a, b) × HCF(a, b) should equal a × b
• If a question gives you LCM and one number, find HCF first, then the other number
• This relationship is tested indirectly in word problems — recognize when to use it
• Works for any two positive integers, whether coprime or not

Common SSC MTS Question Types

SSC MTS typically asks LCM-HCF in three flavors: (1) Direct calculation of LCM or HCF of 2-3 numbers, (2) Word problems about bells, lights, or items dividing equally, (3) Finding numbers given their LCM and HCF. All are straightforward if you know the formulas and the relationship.

• Bells/Lights problem: 'Bells ring every 12, 18, 24 seconds. When do they ring together?' → Find LCM
• Division problem: 'Divide 48 apples and 36 oranges into equal groups. Max group size?' → Find HCF
• Number finding: 'Two numbers have HCF = 5, LCM = 60. Find the numbers' → Use the golden formula
• Always read carefully: 'together' or 'same time' hints LCM; 'divide equally' or 'largest' hints HCF

• Confusing LCM and HCF: Remember — LCM is LARGER (for cycles/together), HCF is SMALLER (for dividing). Use the first letter: L = Large, H = Highest but smaller value.
• Forgetting the golden relationship: Many aspirants calculate LCM and HCF separately instead of using LCM × HCF = a × b. This costs time and invites errors.
• Prime factorization errors: Writing 12 = 2 × 6 instead of 2^2 × 3 leads to wrong LCM/HCF. Always fully factorize into primes.
• Applying the relationship to 3+ numbers: The formula LCM × HCF = a × b works ONLY for two numbers. For three numbers, you must find LCM/HCF step-by-step.
• Misreading word problems: 'Bells ring together' = LCM. 'Divide into equal groups' = HCF. Swapping these ruins the entire solution.

• LCM = Large, Common, Multiple. Think 'Least' as in 'smallest meeting point'. HCF = Highest, Common, Factor. Think 'largest chunk'.
• LCM is for CYCLES (bells, lights, meetings). HCF is for CHUNKS (dividing, packing, grouping).
• Golden Rule: LCM × HCF = a × b. Say it three times before the exam. It's your safety net.
• Prime factorization: LCM takes HIGHEST powers, HCF takes LOWEST powers. HIGH for LCM, LOW for HCF.

• SSC MTS typically gives 1-2 questions on LCM-HCF in the 60-minute quant section. They are usually in the first 20 questions (easier difficulty). Expect 2-3 minutes per question.
• Direct calculation questions (find LCM/HCF of 12, 18, 24) are most common. Use prime factorization for speed. Avoid Euclidean algorithm unless numbers are very large (rare in MTS).
• Word problems are disguised LCM/HCF questions. Always identify: Are items coming together (LCM) or being divided (HCF)? This one insight solves 90% of word problems.
• The golden relationship LCM × HCF = a × b appears in 1-2 indirect questions per paper. If a question feels stuck, try this formula — it often unlocks the answer instantly.
• Recent MTS papers show a trend toward 'find the numbers given LCM and HCF' type questions. Practice these: they test conceptual understanding, not just calculation.