IPMAT HCF and LCM Important Questions with Solutions

IPMAT HCF and LCM important questions are among the most predictable and scoring topics in the Quantitative Aptitude section of the IPMAT (Integrated Programme in Management Aptitude Test) conducted by IIM Indore and IIM Rohtak.

To strengthen your number system preparation, it is helpful to review the complete IPMAT quantitative aptitude syllabus and follow a structured IPMAT preparation strategy, along with regular practice with HCF and LCM questions.

• HCF (Highest Common Factor)is the largest number that divides two or more numbers exactly.
• LCM (Lowest Common Multiple)is the smallest number that is a multiple of two or more numbers.

Why this topic matters for IPMAT:

• It is a high-weightage subtopic within the Number System.
• Questions are formula-driven, so they can be solved quickly with practice.
• It connects directly to Prime Factorization, Factors, Multiples, and Divisibility Rules.
• Most questions are conceptual rather than calculation-heavy, making them ideal for time management.

What you will learn from this article:

• Clear definitions and the core difference between HCF and LCM.
• All important formulas and properties.
• Every major question type with solving methods.
• 25 original solved questions across Easy, Moderate, and Advanced levels.
• Practice sets, common mistakes, a revision sheet, and FAQs.

What Are HCF and LCM?

 HCF is the greatest number that exactly divides two or more numbers, while LCM is the smallest number exactly divisible by those numbers.

These concepts are frequently tested in the Quantitative Aptitude section of the exam. Candidates preparing for the IPMAT 2027 exam should understand HCF and LCM thoroughly, as they are part of the broader Number System syllabus.

• Definition of HCF:The Highest Common Factor is the largest factor shared by two or more numbers. Example: HCF of 12 and 18 is 6.
• Definition of LCM:The Lowest Common Multiple is the smallest number that is a common multiple of two or more numbers. Example: LCM of 12 and 18 is 36.
• Key difference in simple terms:HCF deals with factors (dividing), and LCM deals with multiples (multiplying). HCF ≤ smallest number; LCM ≥ largest number.
• Why important in IPMAT:This topic regularly appears in both the multiple-choice and short-answer sections, and rewards speed because answers follow fixed rules.

HCF vs LCM (Comparison Table)

Aspect	HCF	LCM
Meaning	Largest number dividing all given numbers	Smallest number divisible by all given numbers
Based on	Common factors	Common multiples
Formula usage	HCF = product of lowest powers of common prime factors	LCM = product of highest powers of all prime factors
Size	Always ≤ smallest given number	Always ≥ largest given number
Example (8 & 12)	HCF = 4	LCM = 24
Application in IPMAT	Grouping, max equal distribution, remainder questions	Bells/clocks, repeating events, least-number questions

25 IPMAT HCF and LCM Important Questions with Solutions

These questions cover the HCF and LCM concepts most relevant for IPMAT 2027 preparation.

Easy Level

Q1. Find the HCF of 24 and 36.

• Answer:12
• Solution:24 = 2³×3; 36 = 2²×3². Common = 2²×3 = 12.
• Shortcut: Both are divisible by 12; 12 divides both exactly.
• Exam Tip: For small numbers, scan the largest common divisor mentally.

Q2. Find the LCM of 10 and 15.

• Answer:30
• Solution:10 = 2×5; 15 = 3×5. LCM = 2×3×5 = 30.
• Shortcut: LCM = (10×15)/HCF = 150/5 = 30.
• Exam Tip: Use the product÷HCF rule for two numbers.

Q3. What is the HCF of two co-prime numbers 8 and 9?

• Answer:1
• Solution: Co-prime numbers share no common factor except 1.
• Shortcut: No shared prime factors → HCF = 1.
• Exam Tip: Spot co-primes instantly to save time.

Q4. Find the LCM of 4, 6, and 8.

• Answer:24
• Solution:4 = 2², 6 = 2×3, 8 = 2³. LCM = 2³×3 = 24.
• Shortcut: The highest power of 2 is 2³, of 3 is 3¹.
• Exam Tip:Always pick the highest power of each prime for LCM.

Q5. The HCF of two numbers is 6 and their LCM is 60. If one number is 12, find the other.

• Answer: 30
• Solution: a×b = HCF×LCM → 12×b = 6×60 = 360 → b = 30.
• Shortcut: Other number = (HCF×LCM)/known = 360/12 = 30.
• Exam Tip: Memorise the product relation for instant solving.

Q6. Find the greatest number that divides 18 and 24 exactly.

• Answer:6
• Solution:“Greatest number that divides” = HCF(18,24) = 6.
• Shortcut:18 = 2×3², 24 = 2³×3 → common = 2×3 = 6.
• Exam Tip:“Greatest divides” always signals HCF.

Q7. Find the LCM of 5 and 7.

• Answer:35
• Solution:Both prime and co-prime, so LCM = 5×7 = 35.
• Shortcut:Co-prime → LCM = product.
• Exam Tip:Two distinct primes are always co-prime.

Q8. Two ribbons of 16 m and 20 m are cut into equal pieces of maximum length. Find the length.

• Answer:4 m
• Solution:Maximum equal length = HCF(16,20) = 4.
• Shortcut:Largest common divisor of 16 and 20 is 4.
• Exam Tip:“Maximum equal pieces” → HCF.

Moderate Level

Q9. Find the least number which when divided by 12 and 16 leaves a remainder of 5 each.

• Answer:53
• Solution:LCM(12,16) = 48. Required = 48 + 5 = 53.
• Shortcut:LCM + common remainder.
• Exam Tip:Same remainder added once to the LCM.

Q10. Find the greatest number that divides 64 and 100 leaving remainders 4 each.

• Answer:12
• Solution:64−4 = 60; 100−4 = 96. HCF(60,96): 60 = 2²×3×5, 96 = 2⁵×3 → HCF = 2²×3 = 12.
• Shortcut:Subtract the remainder, then take HCF of the results.
• Exam Tip:Always subtract the remainder before applying HCF.

Q11. The HCF and LCM of two numbers are 8 and 96. If one number is 32, find the other.

• Answer:24
• Solution:Other = (8×96)/32 = 768/32 = 24.
• Shortcut:Product rule for two numbers.
• Exam Tip:Verify HCF divides both results.

Q12. Three bells ring at intervals of 9, 12, and 15 minutes. After how long will they ring together?

• Answer:180 minutes
• Solution:LCM(9,12,15) = 180.
• Shortcut:9 = 3², 12 = 2²×3, 15 = 3×5 → LCM = 2²×3²×5 = 180.
• Exam Tip:“Ring together” → LCM of intervals.

Q13. Find the least number which is exactly divisible by 15, 20, and 25.

• Answer:300
• Solution:15 = 3×5, 20 = 2²×5, 25 = 5². LCM = 2²×3×5² = 300.
• Shortcut:Highest powers: 2², 3, 5².
• Exam Tip:“Exactly divisible by all” → LCM.

Q14. Find the largest number that divides 280 and 1245 leaving remainders 4 and 3 respectively.

• Answer:138
• Solution:280−4 = 276; 1245−3 = 1242. HCF(276,1242) = 138.
• Shortcut:Subtract remainders, then HCF; 276 = 2²×3×23, 1242 = 2×3³×23 → common = 2×3×23 = 138.
• Exam Tip:Match common prime factors carefully.

Q15. Find the HCF of the fractions 2/3, 4/9, and 6/27.

• Answer:2/27
• Solution:HCF of numerators (2,4,6) = 2; LCM of denominators (3,9,27) = 27 → 2/27.
• Shortcut:HCF(numerators)/LCM(denominators).
• Exam Tip:Reverse the rule for LCM of fractions.

Q16. Find the smallest number which when increased by 7 is divisible by 8, 12, and 16.

• Answer:41
• Solution:LCM(8,12,16) = 48. Number = 48 − 7 = 41.
• Shortcut:LCM minus the addition.
• Exam Tip:“Increased by x → divisible” means LCM − x.

Q17. Two numbers are in the ratio 3:4 and their HCF is 5. Find their LCM.

• Answer:60
• Solution:Numbers = 15 and 20. LCM = (15×20)/5 = 60.
• Shortcut:Ratio numbers × HCF = actual numbers.
• Exam Tip:Co-prime ratio terms × HCF give the numbers.

Advanced Level

Q18. The product of two numbers is 2160 and their HCF is 12. Find their LCM.

• Answer:180
• Solution:LCM = product/HCF = 2160/12 = 180.
• Shortcut: Direct division.
• Exam Tip: Product÷HCF is faster than factorization.

Q19. Find the greatest 4-digit number divisible by 15, 25, and 40.

• Answer:9600
• Solution:LCM(15,25,40) = 600. Largest 4-digit multiple: 9999÷600 ≈ 16.6 → 16×600 = 9600.
• Shortcut:Floor(9999/LCM) × LCM.
• Exam Tip:Use the floor of (limit ÷ LCM).

Q20. Find the least number which when divided by 6, 9, and 12 leaves remainders 2, 5, and 8 respectively.

• Answer:32
• Solution:Difference (divisor − remainder) is constant: 6−2 = 9−5 = 12−8 = 4. LCM(6,9,12) = 36. Number = 36 − 4 = 32.
• Shortcut:When divisor − remainder is constant, answer = LCM − that constant.
• Exam Tip:Check the constant gap before solving.

Q21. Find the smallest number which leaves remainder 4 when divided by 5, 6, and 7.

• Answer:214
• Solution:LCM(5,6,7) = 210. Number = 210 + 4 = 214.
• Shortcut:LCM + common remainder.
• Exam Tip:Validate that the remainder is smaller than each divisor.

Q22. Three numbers are in the ratio 2:3:4 and their LCM is 240. Find the numbers.

• Answer:40, 60, 80
• Solution:Let numbers be 2x, 3x, 4x. LCM = 12x = 240 → x = 20. Numbers = 40, 60, 80.
• Shortcut:LCM of ratio terms × x.
• Exam Tip:Compute LCM of ratio terms first.

Q23. Find the greatest number that divides 1657 and 2037 leaving remainders 6 and 5 respectively.

• Answer:127
• Solution:1657−6 = 1651; 2037−5 = 2032. HCF(1651,2032) = 127.
• Shortcut:1651 = 13×127, 2032 = 16×127 → HCF = 127.
• Exam Tip:Look for a shared large factor.

Q24. A number when divided by 5, 6, 7, and 8 leaves remainder 3 in each case, but is exactly divisible by 9. Find the smallest such number.

• Answer:1683
• Solution:LCM(5,6,7,8) = 840. Number = 840k + 3 divisible by 9. Test k: 840×2+3 = 1683; 1683÷9 = 187. ✔
• Shortcut:Form 840k + 3 and test divisibility by 9.
• Exam Tip:Combine LCM with an extra divisibility condition.

Q25. The HCF of two numbers is 11 and their LCM is 7700. If one number is 275, find the other.

• Answer:308
• Solution:Other = (11×7700)/275 = 84700/275 = 308.
• Shortcut:Product rule.
• Exam Tip:Confirm HCF (11) divides the answer (308 = 11×28). 

Important Formulas and Properties

The most-used rule is HCF × LCM = Product of the two numbers.

• HCF × LCM relation (two numbers):
  • HCF(a, b) × LCM(a, b) = a × b
  • This works only for two numbers, not three or more.
• Co-prime rule:
  • Two numbers are co-primeif their HCF = 1.
  • For co-prime numbers, LCM = product of the numbers.
• Remainder-based properties:
  • Greatest number that divides a, b, c leaving the same remainder= HCF of differences (b−a), (c−b), (c−a).
  • Greatest number dividing a, b, c leaving remainders r₁, r₂, r₃ = HCF of (a−r₁), (b−r₂), (c−r₃).
  • Least number leaving remainder r when divided by a, b, c = (LCM of a, b, c) + r.
• Fraction-based rules:
  • HCF of fractions = HCF of numerators ÷ LCM of denominators.
  • LCM of fractions = LCM of numerators ÷ HCF of denominators.

Types of IPMAT HCF and LCM Questions

Direct HCF Questions

• What it is: Find the HCF of given numbers directly.
• How to identify it: Keywords like “greatest number that divides” or “maximum equal groups.”
• Step-by-step method:
  • Break each number into prime factors.
  • Pick common primes with their lowest powers.
  • Multiply them.
• Example: HCF of 48 and 60 → 48 = 2⁴×3, 60 = 2²×3×5 → HCF = 2²×3 = 12.

Direct LCM Questions

• What it is: Find the LCM of given numbers directly.
• How to identify it: Keywords like “smallest number divisible by” or “events repeating together.”
• Step-by-step method:
  • Prime factorize each number.
  • Take all primes with their highest powers.
  • Multiply them.
• Example: LCM of 9 and 12 → 9 = 3², 12 = 2²×3 → LCM = 2²×3² = 36.

Remainder-Based Questions

• What it is: Numbers that leave specific remainders on division.
• How to identify it: Phrases like “leaves remainder 3” or “leaves the same remainder.”
• Step-by-step method:
  • For same remainder:  take HCF of the differences.
  • For least number: compute LCM, then add the remainder.
• Example: Least number leaving remainder 2 when divided by 4 and 6 → LCM(4,6)=12 → 12 + 2 = 14.

Word Problems

• What it is: Real-life framing of HCF or LCM (tiles, ribbons, distribution).
• How to identify it:“Maximum equal pieces” → HCF; “minimum length/quantity” → LCM.
• Step-by-step method:
  • Decide HCF (dividing into equal parts) or LCM (finding common point).
  • Apply the relevant formula.
• Example: Largest tape that measures 36 cm and 48 cm exactly → HCF(36,48) = 12 cm.

Bells and Clocks Problems

• What it is: Events (bells, lights, signals) that recur at fixed intervals.
• How to identify it: “Ring/toll together,” “after how many seconds again.”
• Step-by-step method:
  • Find the LCM of the intervals → the time they coincide.
  • Divide the total time by LCM to count occurrences.
• Example: Bells ring every 6 and 8 seconds → ring together every LCM(6,8) = 24 seconds.

Greatest/Least Number Problems

• What it is: Finding the greatest or least number satisfying divisibility conditions.
• How to identify it: “Greatest number that divides...” (HCF) or “least number divisible by...” (LCM).
• Step-by-step method:
  • Greatest → HCF route.
  • Least → LCM route, adjusting for remainders.
• Example: Greatest number dividing 70 and 125, leaving remainders 5 and 8 → HCF(65,117) = 13.

High-Frequency Exam Patterns

Common patterns in IPMAT HCF and LCM questions:

• Product relation problems: Given HCF, LCM, and one number → find the other. Frequently asked because they test the core formula quickly.
• Remainder-based HCF problems: “Greatest number leaving remainder r.” Common because they combine subtraction logic with HCF.
• Bells/clocks LCM problems: “Events occurring together.” Popular as application-based questions.
• Least-number LCM problems: “Smallest number divisible by...” Tests the highest-power logic.
• Ratio-based HCF/LCM problems: Numbers given as ratios. Tests conversion to actual values.

How to identify them quickly:

• “Greatest / maximum / largest divides” → HCF.
• “Least / smallest / together / repeats” → LCM.
• “Leaves remainder” → adjust before applying HCF or LCM.

IPMAT HCF and LCM Practice Questions

Easy Practice Set

• Find the HCF of 28 and 42.
• Find the LCM of 9 and 12.
• Find the HCF of co-prime numbers 14 and 15.
• Find the smallest number divisible by 6, 8, and 10.
• The HCF of two numbers is 4, LCM is 48, one number is 16 — find the other.

Moderate Practice Set

• Find the least number leaving remainder 3 when divided by 10 and 15.
• Find the greatest number dividing 65 and 117 exactly.
• Three bells ring at 8, 12, and 16 seconds — when do they ring together?
• Two numbers in ratio 5:6 have HCF 7 — find their LCM.
• Find the HCF of fractions 3/4, 6/8, and 9/16.

Advanced Practice Set

• Find the greatest 3-digit number divisible by 12, 15, and 18.
• A number leaves remainder 5 when divided by 7, 8, and 9 — find the smallest.
• Three numbers in ratio 3:4:5 have LCM 360 — find the numbers.
• Find the greatest number dividing 2011 and 2401 leaving remainders 9 and 7.
• The product of two numbers is 4032, HCF is 12 — find the LCM.

Common Mistakes Students Make

Mistake	Correct Approach
Using lowest powers for LCM	Use highest powers of all primes for LCM
Using highest powers for HCF	Use lowest powers of common primes for HCF
Applying HCF×LCM = product to 3+ numbers	This rule works only for two numbers
Forgetting to subtract remainders before HCF	Always subtract remainders, then take HCF
Adding remainder before finding LCM	Find LCM first, then add the remainder
Confusing “greatest divides” with LCM	“Greatest divides” → HCF; “least divisible” → LCM
Treating all numbers as co-prime	Verify HCF = 1 before using LCM = product

Quick Revision Sheet

Key Formulas:

• HCF × LCM = product of two numbers.
• LCM of fractions = LCM(numerators)/HCF(denominators).
• HCF of fractions = HCF(numerators)/LCM(denominators).
• Co-prime → LCM = product, HCF = 1.

Key Properties:

• HCF ≤ smallest number; LCM ≥ largest number.
• Greatest number leaving same remainder = HCF of differences.
• Least number leaving remainder r = LCM + r.

Important Shortcuts:

• Two numbers, find missing → (HCF×LCM)/known.
• “Increased by x, divisible” → LCM − x.
• Constant (divisor − remainder) gap → LCM − constant.

Exam-Day Tips:

• Identify HCF vs LCM from keywords first.
• Use prime factorization for accuracy, product rule for speed.
• Always verify HCF divides your final answer.

Mastering IPMAT HCF and LCM important questions is one of the most efficient ways to boost your Quantitative Aptitude score for IIM Indore and IIM Rohtak. By understanding core definitions, memorizing the product relation, recognizing keyword patterns, and practicing the HCF and LCM important questions for IPMAT covered above, you can solve these problems with speed and confidence. Use the revision sheet and practice sets regularly, and these IPMAT HCF LCM questions will become guaranteed marks on exam day.