LCM & HCF – Complete Tutorial (Competitive Exams)

COMPLETE MASTERCLASS: LCM AND HCF

1. INTRODUCTION

Definitions

• HCF (Highest Common Factor): Also known as GCD (Greatest Common Divisor). The HCF of two or more given numbers is the largest number that divides each of them completely (leaving no remainder).
• LCM (Least Common Multiple): The LCM of two or more numbers is the smallest number which is perfectly divisible by each of the given numbers.

Importance in Competitive Exams

• Direct Weightage: 2–3 questions are almost guaranteed in Prelims and Mains.
• Foundational Tool: You cannot quickly solve Time and Work, Pipes and Cisterns, or Speed, Time, and Distance without mastering fast LCM calculations.

Real-Life Applications

• LCM: Finding when multiple events will happen together again (e.g., bells ringing together, athletes meeting at the starting point on a circular track, traffic lights changing).
• HCF: Dividing things into equal groups, finding the maximum length of a measuring tape, or tiling a floor with the minimum number of square tiles.

2. CONCEPT THEORY (DETAILED)

Methods to Find HCF

• Prime Factorization Method: Express each number as a product of prime factors. The HCF is the product of the lowest powers of common prime factors.
  Example: HCF of 24 and 36.
  24 = 2³ × 3¹
  36 = 2² × 3²
  HCF = 2² × 3¹ = 12
• Division Method (Useful for large numbers): Divide the larger number by the smaller number. Then, divide the previous divisor by the remainder. Repeat until the remainder is 0. The last divisor is the HCF.

Methods to Find LCM

• Prime Factorization Method: Express each number as a product of prime factors. The LCM is the product of the highest powers of all prime factors involved.
  Example: LCM of 24 and 36.
  24 = 2³ × 3¹
  36 = 2² × 3²
  LCM = 2³ × 3² = 72
• Common Division Method: Write numbers in a row, divide by the smallest prime number that divides at least two of them, carry forward the undivided numbers, and repeat until all quotients are 1. Multiply all divisors.

Important Relationships & Formulas

• Core Formula: LCM × HCF = First Number × Second Number (Note: This is valid ONLY for two numbers).
• Co-prime Numbers: Two numbers are co-prime if their HCF is 1. Their LCM is simply their product.
• Fractions:
  • HCF of fractions = (HCF of Numerators) / (LCM of Denominators)
  • LCM of fractions = (LCM of Numerators) / (HCF of Denominators)

Properties of LCM and HCF

• The HCF of given numbers ALWAYS perfectly divides their LCM.
• The HCF is always less than or equal to the smallest of the given numbers.
• The LCM is always greater than or equal to the largest of the given numbers.

3. TYPES OF QUESTIONS & 4. SOLVED EXAMPLES

Here are 40 highly relevant solved examples covering every pattern asked in SSC, Banking, and Railway exams.

Type A: Basic Calculation & Fractions

Q1. Find the HCF of 84, 126, and 140.

View Solution

Find the difference between the closest numbers: 140 - 126 = 14. The HCF will be 14 or its factor. 14 divides 84, 126, and 140.
Answer: 14.

Q2. Find the LCM of 15, 24, 32, and 45.

View Solution

Take the largest (45). Prime factors: 45 = 3² × 5. 32 = 2⁵. 24 = 2³ × 3. 15 = 3 × 5. LCM must have the highest powers: 2⁵ × 3² × 5 = 32 × 9 × 5.
Answer: 1440.

Q3. Find the HCF of 2/3, 4/5, and 6/7.

View Solution

HCF = (HCF of 2, 4, 6) / (LCM of 3, 5, 7) = 2/105.
Answer: 2/105.

Q4. Find the LCM of 3/4, 6/7, and 9/8.

View Solution

LCM = (LCM of 3, 6, 9) / (HCF of 4, 7, 8) = 18/1.
Answer: 18.

Q5. Find the HCF of 1.2, 0.24, and 6.

View Solution

Convert to whole numbers by multiplying by 100: 120, 24, 600. HCF of (120, 24, 600) is 24. Divide by 100 again.
Answer: 0.24.

Q6. Find the LCM of 0.6, 9.6, and 0.36.

View Solution

Multiply by 100: 60, 960, 36. LCM of (60, 960, 36) = 2880. Divide by 100 = 28.8.
Answer: 28.8.

Type B: Ratio-Based Problems

Q7. Two numbers are in the ratio 3:4. Their HCF is 15. Find the numbers.

View Solution

Numbers are 3 × 15 and 4 × 15.
Answer: 45 and 60.

Q8. The ratio of two numbers is 4:5 and their LCM is 120. Find the numbers.

View Solution

Let numbers be 4x and 5x. Since they are co-prime, LCM = 4 × 5 × x = 20x. 20x = 120 ⇒ x = 6.
Answer: 24 and 30.

Q9. Three numbers are in the ratio 2:3:4 and their HCF is 12. Find the LCM.

View Solution

Numbers are 2 × 12, 3 × 12, 4 × 12 ⇒ 24, 36, 48. LCM of 24, 36, 48 = 144.
Answer: 144.

Q10. The sum of two numbers is 528 and their HCF is 33. How many such pairs exist?

View Solution

Let numbers be 33a and 33b (where a, b are co-prime). 33a + 33b = 528 ⇒ a + b = 16. Co-prime pairs summing to 16: (1,15), (3,13), (5,11), (7,9).
Answer: 4 pairs.

Q11. The product of two numbers is 2028 and their HCF is 13. Find the number of pairs.

View Solution

Let numbers be 13a and 13b. 13a × 13b = 2028 ⇒ 169ab = 2028 ⇒ ab = 12. Co-prime pairs yielding 12: (1,12) and (3,4).
Answer: 2 pairs.

Type C: LCM-HCF Relationship

Q12. The LCM of two numbers is 2079 and their HCF is 27. If one number is 189, find the other.

View Solution

LCM × HCF = N1 × N2 ⇒ 2079 × 27 = 189 × N2 ⇒ N2 = 297.
Answer: 297.

Q13. The HCF of two numbers is 11 and their LCM is 7700. If one of the numbers lies between 275 and 300, find the numbers.

View Solution

N1 × N2 = 11 × 7700. Let numbers be 11a, 11b. 11a × 11b = 11 × 7700 ⇒ ab = 700. Co-prime pairs: a=25, b=28. Numbers: 11 × 25 = 275 and 11 × 28 = 308.
Answer: 275 and 308.

Q14. The LCM of two numbers is 44 times their HCF. The sum of LCM and HCF is 1125. If one number is 25, find the other.

View Solution

L = 44H. L + H = 1125 ⇒ 45H = 1125 ⇒ H = 25. L = 1100. 25 × 1100 = 25 × N2 ⇒ N2 = 1100.
Answer: 1100.

Q15. Can two numbers have 16 as their HCF and 380 as their LCM?

View Solution

HCF must exactly divide LCM. 380 / 16 = 23.75 (Not an integer).
Answer: No.

Type D: LCM Remainder Problems

Rule 1: "Find the least number which when divided by x, y, z leaves same remainder R in each case." → Form: LCM(x,y,z) × k + R

Q16. Find the least number which when divided by 12, 15, 20, and 54 leaves a remainder of 8 in each case.

View Solution

LCM(12, 15, 20, 54) = 540. Number = 540 + 8.
Answer: 548.

Q17. Find the least number which when divided by 5, 6, 7, and 8 leaves a remainder 3, but when divided by 9 leaves no remainder.

View Solution

LCM(5, 6, 7, 8) = 840. Number is of the form 840k + 3. For k=1, 843 (not divisible by 9). For k=2, 1683 (divisible by 9).
Answer: 1683.

Rule 2: "Find the least number which when divided by x, y, z leaves remainders a, b, c respectively." → If (x-a) = (y-b) = (z-c) = K, then Form: LCM(x,y,z) - K

Q18. Find the least number which when divided by 20, 25, 35, and 40 leaves remainders 14, 19, 29, and 34 respectively.

View Solution

Notice the difference: 20-14=6, 25-19=6, etc. K=6. LCM(20, 25, 35, 40) = 1400. Number = 1400 - 6.
Answer: 1394.

Q19. Find the greatest 4-digit number exactly divisible by 12, 15, 20, and 35.

View Solution

LCM(12, 15, 20, 35) = 420. Greatest 4-digit number = 9999. 9999 / 420 leaves remainder 339. Number = 9999 - 339.
Answer: 9660.

Q20. Find the greatest 5-digit number which when divided by 3, 5, 8, 12 leaves remainder 2.

View Solution

LCM(3, 5, 8, 12) = 120. 99999 / 120 ⇒ Remainder is 39. Number divisible = 99999 - 39 = 99960. Add remainder 2.
Answer: 99962.

Type E: HCF Remainder Problems

Rule 1: "Find the greatest number that divides x, y, z leaving same remainder R." → Form: HCF of (x-R, y-R, z-R)

Q21. Find the greatest number that will divide 43, 91, and 183 so as to leave the same remainder in each case.

View Solution

When remainder is unknown, take the HCF of their absolute differences: |91-43|, |183-91|, |183-43| = 48, 92, 140. HCF of (48, 92, 140).
Answer: 4.

Q22. Find the greatest number that divides 1305, 4665, and 6905 leaving the same remainder in each case.

View Solution

Differences: 4665 - 1305 = 3360; 6905 - 4665 = 2240; 6905 - 1305 = 5600. HCF(3360, 2240, 5600).
Answer: 1120.

Rule 2: "Find the greatest number that divides x, y, z leaving remainders a, b, c." → Form: HCF of (x-a, y-b, z-c)

Q23. Find the greatest number which divides 29, 60, and 103 leaving remainders 5, 12, and 7 respectively.

View Solution

29 - 5 = 24; 60 - 12 = 48; 103 - 7 = 96. HCF of (24, 48, 96).
Answer: 24.

Q24. What is the greatest number that will divide 99, 123, and 183 leaving the same remainder 3 in each case?

View Solution

99 - 3 = 96; 123 - 3 = 120; 183 - 3 = 180. HCF of (96, 120, 180).
Answer: 24.

Type F: Real-life Word Problems

Q25. Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10, and 12 seconds respectively. In 30 minutes, how many times do they toll together?

View Solution

LCM(2, 4, 6, 8, 10, 12) = 120 seconds = 2 minutes. They toll together every 2 mins. In 30 mins: (30 / 2) + 1 (for the 0th minute) = 16 times.
Answer: 16 times.

Q26. Three persons walking around a circular track of 11 km. A speeds at 4 kmph, B at 5.5 kmph, C at 8 kmph. When will they meet at the starting point?

View Solution

Time = Distance / Speed. Times are 11/4, 11/5.5, 11/8 hours ⇒ 11/4, 2/1, 11/8. LCM of fractions = LCM(11, 2, 11) / HCF(4, 1, 8) = 22 / 1.
Answer: 22 hours.

Q27. 3 traffic lights change after 48 sec, 72 sec, and 108 sec. If they all change simultaneously at 8:20:00 hours, then at what time will they change together again?

View Solution

LCM(48, 72, 108) = 432 seconds = 7 mins 12 secs.
Answer: 8:27:12 hrs.

Q28. A gardener has 44 apple trees, 66 banana trees, and 110 mango trees. He wants to plant them in rows such that each row has the same number of trees of only one type. Minimum number of rows?

View Solution

Trees per row = HCF(44, 66, 110) = 22. Total rows = (44/22) + (66/22) + (110/22) = 2 + 3 + 5.
Answer: 10 rows.

Q29. A rectangular courtyard 3.78 m long and 5.25 m wide is to be paved exactly with square tiles, all of the same size. Find the largest size of the tile.

View Solution

Convert to cm: 378 cm, 525 cm. HCF(378, 525).
Answer: 21 cm.

Q30. From the above question (Q29), find the minimum number of tiles required.

View Solution

Total Tiles = Area of floor / Area of one tile = (378 × 525) / (21 × 21) = 18 × 25.
Answer: 450.

Type G: Algebra & Polynomials

Q31. Find the HCF of a²b⁴c⁶, b³c⁸a⁴, and a⁸b⁶c².

View Solution

Take the lowest power of each variable.
Answer: a²b³c².

Q32. Find the LCM of the same terms: a²b⁴c⁶, b³c⁸a⁴, and a⁸b⁶c².

View Solution

Take the highest power of each variable.
Answer: a⁸b⁶c⁸.

Q33. Find the HCF of (x² - 4) and (x² + 4x + 4).

View Solution

x² - 4 = (x-2)(x+2). x² + 4x + 4 = (x+2)². Common term: (x+2).
Answer: x+2.

Q34. Find the LCM of (x³ - y³) and (x² - y²).

View Solution

(x-y)(x²+xy+y²) and (x-y)(x+y). Highest powers:
Answer: (x-y)(x+y)(x²+xy+y²).

Type H: Advanced Miscellaneous

Q35. The sum of two numbers is 36 and their HCF and LCM are 3 and 105 respectively. The sum of the reciprocals of two numbers is?

View Solution

Let numbers be a, b. a+b = 36. ab = H × L = 3 × 105 = 315. Reciprocal sum = 1/a + 1/b = (a+b)/ab = 36/315.
Answer: 4/35.

Q36. Find the least perfect square exactly divisible by 3, 4, 5, 6, and 8.

View Solution

LCM(3, 4, 5, 6, 8) = 120. Prime factors of 120 = 2³ × 3 × 5. To make it a perfect square, multiply by 2 × 3 × 5 = 30. 120 × 30.
Answer: 3600.

Q37. The HCF and LCM of two numbers are 21 and 4641. If one number lies between 200 and 300, find the numbers.

View Solution

21a × 21b = 21 × 4641 ⇒ ab = 221. Co-prime factors of 221 are 13 and 17. Numbers: 21 × 13 = 273 and 21 × 17 = 357.
Answer: 273.

Q38. HCF of 3⁴⁵ - 1 and 3³⁵ - 1.

View Solution

Rule: HCF of (A^m - 1) and (Aⁿ - 1) = A^{HCF of (m,n)} - 1. HCF of 45, 35 = 5.
Answer: 3⁵ - 1 = 242.

Q39. What is the greatest number that will divide 38, 45, and 52 and leave remainders 2, 3, and 4 respectively?

View Solution

38 - 2 = 36; 45 - 3 = 42; 52 - 4 = 48. HCF(36, 42, 48).
Answer: 6.

Q40. Product of two co-prime numbers is 117. What is their LCM?

View Solution

For co-prime numbers, HCF is 1. Therefore, LCM = Product.
Answer: 117.

5. SHORTCUTS & TRICKS

• The Difference Trick (HCF): The HCF of any two numbers NEVER exceeds their difference. It will either be the difference itself or a factor of the difference. (Example: HCF of 68 and 85. Difference = 17. 17 divides both. Boom! HCF is 17.)
• The Largest Number Trick (LCM): To find the LCM, take the largest number and check if the other numbers divide it. If not, multiply the largest number by 2, 3, 4... until you find a multiple that is divisible by the other numbers. (Example: LCM of 4, 6, 8. Largest is 8. 6 doesn't divide 8. Try 8 × 2 = 16 (no). Try 8 × 3 = 24. 4 and 6 divide 24. LCM is 24.)
• Divisibility Rule Bypass: For questions asking "Find the least number divisible by 9, 12, 15, 18", just look at the options! The correct option must satisfy the divisibility rule for 9 (sum of digits is a multiple of 9).

6. PRACTICE SET

Try these 30 questions independently. Click "Reveal Answer" to check your work!

1. Find the HCF of 36 and 84.

Reveal Answer

Answer: 12

2. Find the LCM of 12, 18, 24.

Reveal Answer

Answer: 72

3. Find the HCF of 3/5, 9/10, 15/16.

Reveal Answer

Answer: 3/80

4. Find the LCM of 2/3, 4/9, 5/6.

Reveal Answer

Answer: 20/3

5. Two numbers are in ratio 5:6. If their HCF is 4, what is the LCM?

Reveal Answer

Answer: 120

6. The product of two numbers is 1280 and their HCF is 8. Find their LCM.

Reveal Answer

Answer: 160

7. Find the greatest number that divides 122 and 243 leaving remainders 2 and 3 respectively.

Reveal Answer

Answer: 120

8. Find the least number divisible by 12, 15, and 20 leaving remainder 4 in each case.

Reveal Answer

Answer: 64

9. Three bells toll at intervals of 15, 20, and 30 minutes. If they toll together at 10:00 AM, when will they toll together next?

Reveal Answer

Answer: 11:00 AM

10. The HCF of two numbers is 8. Which of the following CANNOT be their LCM? (a) 24 (b) 48 (c) 56 (d) 60

Reveal Answer

Answer: 60 (Not divisible by 8)

11. Sum of two numbers is 100, HCF is 5. How many such pairs exist?

Reveal Answer

Answer: 4 pairs

12. Find the greatest number dividing 43, 91, 183 leaving same remainder.

Reveal Answer

Answer: 4

13. HCF of 0.54, 1.8, and 7.2.

Reveal Answer

Answer: 0.18

14. LCM of two numbers is 1920, HCF is 16. One number is 128, find the other.

Reveal Answer

Answer: 240

15. A number divided by 10 leaves 9, by 9 leaves 8, by 8 leaves 7. Find the least such number.

Reveal Answer

Answer: 359 (LCM of 10,9,8 - 1)

16. HCF of x³y²z and x²y³z².

Reveal Answer

Answer: x²y²z

17. Minimum tiles required for a floor 15m 17cm long and 9m 2cm broad.

Reveal Answer

Answer: 814

18. Least perfect square divisible by 21, 36, 66.

Reveal Answer

Answer: 213444

19. Sum of HCF and LCM of two numbers is 680, LCM is 84 times HCF. One number is 56, find the other.

Reveal Answer

Answer: 96

20. Greatest 4-digit number divisible by 15, 25, 40, 75.

Reveal Answer

Answer: 9600

21. HCF of two co-prime numbers.

Reveal Answer

Answer: 1

22. Product of two co-prime numbers is 143. Find their LCM.

Reveal Answer

Answer: 143

23. The LCM of two numbers is 12 times their HCF. The sum of HCF and LCM is 403. If both numbers are less than LCM, find the numbers.

Reveal Answer

Answer: 93 and 124

24. Find the least number completely divisible by 16, 24, 36, 54.

Reveal Answer

Answer: 432

25. 4 traffic lights change at 30, 45, 60, 120 seconds. They changed together at 1:00 PM. Next time?

Reveal Answer

Answer: 1:06 PM

26. HCF of 2¹⁰⁰ - 1 and 2¹²⁰ - 1.

Reveal Answer

Answer: 2²⁰-1

27. Ratio of two numbers is 3:4. Their LCM is 84. Find the larger number.

Reveal Answer

Answer: 28

28. Find the least number which when divided by 5, 6, 7 leaves remainder 3, but is exactly divisible by 9.

Reveal Answer

Answer: 1683

29. Two numbers have 15 as HCF and 300 as LCM. Can this exist?

Reveal Answer

Answer: Yes (300/15 = 20)

30. The LCM of 3, 2.7, and 0.09.

Reveal Answer

Answer: 27

7. EXAM STRATEGY

How to Identify Question Types Quickly

• Keyword "Greatest / Maximum / Largest / Longest": Almost always implies finding the HCF.
• Keyword "Least / Minimum / Smallest / Together again": Almost always implies finding the LCM.
• Circular Track / Tolling Bells / Traffic Lights: This is a classic LCM of the time intervals. Always ensure the units (seconds/minutes/hours) are identical before calculating.

Time-Saving Tips (The "Option Elimination" Method)

For competitive exams (SSC/Banking), do not solve LCM/HCF questions traditionally if you can avoid it. Use Divisibility Rules: If a question asks for a number divisible by 12, 15, and 18, check the options. The correct answer MUST be divisible by 9 (since 18 = 9×2). Use the rule of 9 (sum of digits divisible by 9) to eliminate wrong options instantly in 2 seconds.

Common Mistakes to Avoid

• Forgetting to add/subtract the remainder: In LCM remainder problems, memorize whether to Add or Subtract. If it's the same remainder, ADD. If it's different remainders, find the constant difference (K) and SUBTRACT.
• Mixing up Fraction Formulas: Remember, whatever you are asked to find (LCM or HCF of the fraction), do THAT to the numerator. The opposite happens to the denominator.
• Ignoring Units: In word problems, speeds might be in km/hr but lengths in meters. Always standardize units before finding the LCM.