Episode 8 — Aptitude and Reasoning / 8.7 — HCF and LCM

8.7 Practice MCQs

Instructions: Choose the best answer for each question. Answers with detailed explanations follow each question.

Easy (Questions 1--15)

Q1. Find the HCF of 36 and 84.

(a) 6 (b) 12 (c) 18 (d) 24

Answer: (b) 12

36 = 2^2 x 3^2
84 = 2^2 x 3 x 7
HCF = 2^2 x 3 = 12

Q2. Find the LCM of 9 and 12.

(a) 24 (b) 36 (c) 48 (d) 72

Answer: (b) 36

9 = 3^2
12 = 2^2 x 3
LCM = 2^2 x 3^2 = 4 x 9 = 36

Q3. The HCF of two numbers is 4 and their LCM is 48. If one number is 16, the other is:

(a) 8 (b) 12 (c) 24 (d) 36

Answer: (b) 12

Other number = (HCF x LCM) / given number = (4 x 48) / 16 = 192 / 16 = 12

Q4. HCF of two consecutive even numbers is always:

(a) 1 (b) 2 (c) 4 (d) The smaller number

Answer: (b) 2

Consecutive even numbers differ by 2.
Example: HCF(14, 16) = 2. HCF(100, 102) = 2. Always 2.

Q5. If the HCF of two numbers is 1, the numbers are called:

(a) Twin primes (b) Co-prime (c) Composite (d) Perfect

Answer: (b) Co-prime

By definition, numbers with HCF = 1 are co-prime (relatively prime).
Note: They need not be individually prime. Example: 8 and 15 are co-prime.

Q6. Find the LCM of 4, 6, and 8.

(a) 12 (b) 24 (c) 48 (d) 96

Answer: (b) 24

4 = 2^2
6 = 2 x 3
8 = 2^3
LCM = 2^3 x 3 = 24

Q7. The product of two numbers is 1680 and their HCF is 14. Their LCM is:

(a) 60 (b) 100 (c) 120 (d) 140

Answer: (c) 120

LCM = Product / HCF = 1680 / 14 = 120

Q8. Find the HCF of 2/3 and 4/5.

(a) 2/15 (b) 4/15 (c) 2/5 (d) 4/3

Answer: (a) 2/15

HCF of numerators: HCF(2, 4) = 2
LCM of denominators: LCM(3, 5) = 15
HCF = 2/15

Q9. LCM of two co-prime numbers 7 and 11 is:

(a) 7 (b) 11 (c) 77 (d) 1

Answer: (c) 77

For co-prime numbers, LCM = product = 7 x 11 = 77.

Q10. Which of the following is true?

(a) HCF is always greater than LCM (b) LCM is always greater than HCF (c) HCF always divides LCM (d) Both (b) and (c)

Answer: (d) Both (b) and (c)

LCM >= max(a, b) >= min(a, b) >= HCF.
Also, HCF always divides LCM (a fundamental property).
When a = b, HCF = LCM = a (so LCM >= HCF, and both hold).

Q11. Find the HCF of 72 and 108.

(a) 12 (b) 18 (c) 36 (d) 54

Answer: (c) 36

72  = 2^3 x 3^2
108 = 2^2 x 3^3
HCF = 2^2 x 3^2 = 4 x 9 = 36

Q12. If LCM(a, b) = b, which of the following must be true?

(a) a = b (b) a divides b (c) b divides a (d) a and b are co-prime

Answer: (b) a divides b

LCM(a, b) = b means b is already a multiple of a. So a divides b.
Example: LCM(5, 15) = 15, and 5 divides 15.

Q13. LCM of 15, 20, and 25 is:

(a) 100 (b) 200 (c) 300 (d) 600

Answer: (c) 300

15 = 3 x 5
20 = 2^2 x 5
25 = 5^2
LCM = 2^2 x 3 x 5^2 = 4 x 3 x 25 = 300

Q14. Find the HCF of 18, 42, and 60.

(a) 2 (b) 3 (c) 6 (d) 12

Answer: (c) 6

18 = 2 x 3^2
42 = 2 x 3 x 7
60 = 2^2 x 3 x 5
HCF = 2 x 3 = 6

Q15. If HCF(a, b) = a, then:

(a) a = b (b) a is a multiple of b (c) a is a factor of b (d) a and b are co-prime

Answer: (c) a is a factor of b

HCF(a, b) = a means a divides both numbers.
Since a divides b, a is a factor of b.
Also LCM(a, b) = b in this case.

Medium (Questions 16--30)

Q16. Three bells toll at intervals of 9, 12, and 15 minutes. If they start together, after how many minutes do they toll together again?

(a) 60 (b) 120 (c) 180 (d) 360

Answer: (c) 180

LCM(9, 12, 15):
  9 = 3^2
  12 = 2^2 x 3
  15 = 3 x 5
  LCM = 2^2 x 3^2 x 5 = 4 x 9 x 5 = 180 minutes

Q17. The HCF of two numbers is 23 and the other two factors of their LCM are 13 and 14. The larger number is:

(a) 276 (b) 299 (c) 322 (d) 345

Answer: (c) 322

The two numbers are 23 x 13 = 299 and 23 x 14 = 322.
(Here 13 and 14 are co-prime, so this is valid.)
The larger number is 322.

Q18. Find the largest number that divides 62, 132, and 237 leaving the same remainder.

(a) 35 (b) 30 (c) 25 (d) 20

Answer: (a) 35

Differences: 132 - 62 = 70,  237 - 132 = 105,  237 - 62 = 175
HCF(70, 105) = 35
Check: HCF(35, 175) = 35
Verification: 62/35 = 1 R 27; 132/35 = 3 R 27; 237/35 = 6 R 27. Same remainder.

Q19. The smallest number which when divided by 12, 15, and 20 leaves a remainder of 5 in each case is:

(a) 55 (b) 60 (c) 65 (d) 75

Answer: (c) 65

LCM(12, 15, 20) = 60
Required number = 60 + 5 = 65
Verification: 65/12 = 5 R 5; 65/15 = 4 R 5; 65/20 = 3 R 5.

Q20. A floor is 6m 60cm long and 4m 20cm wide. Find the minimum number of square tiles required to cover the floor.

(a) 77 (b) 350 (c) 462 (d) 154

Answer: (a) 77

Length = 660 cm, Width = 420 cm
Largest tile side = HCF(660, 420) = 60 cm

Number of tiles = (660/60) x (420/60) = 11 x 7 = 77

Q21. The LCM of two numbers is 4 times their HCF. The sum of LCM and HCF is 125. Find the product of the two numbers.

(a) 500 (b) 1500 (c) 2500 (d) 3125

Answer: (c) 2500

LCM = 4 x HCF
LCM + HCF = 125
4h + h = 125 --> 5h = 125 --> h = 25
LCM = 100

Product = HCF x LCM = 25 x 100 = 2500

Q22. How many pairs of positive integers have HCF = 5 and LCM = 105?

(a) 1 (b) 2 (c) 3 (d) 4

Answer: (b) 2

x * y = 105 / 5 = 21, where HCF(x, y) = 1

Co-prime factor pairs of 21:
  (1, 21): HCF = 1 --> (5, 105)
  (3, 7):  HCF = 1 --> (15, 35)

2 pairs.

Q23. Find the smallest number which when divided by 6, 9, and 12 leaves no remainder.

(a) 18 (b) 24 (c) 36 (d) 72

Answer: (c) 36

LCM(6, 9, 12):
  6 = 2 x 3
  9 = 3^2
  12 = 2^2 x 3
  LCM = 2^2 x 3^2 = 36

Q24. The HCF of two numbers is 15 and their LCM is 300. If one number is 60, find the other.

(a) 45 (b) 50 (c) 75 (d) 100

Answer: (c) 75

Other = (HCF x LCM) / 60 = (15 x 300) / 60 = 4500 / 60 = 75
Verification: HCF(60, 75) = 15. LCM = (60 x 75)/15 = 300. Correct.

Q25. Find the LCM of 2/3, 5/6, and 4/9.

(a) 20/3 (b) 10/3 (c) 20/9 (d) 40/3

Answer: (a) 20/3

LCM of numerators: LCM(2, 5, 4) = 20
HCF of denominators: HCF(3, 6, 9) = 3
LCM = 20/3

Q26. Three runners start at the same time from the same point on a circular track. They take 42, 56, and 63 seconds to complete one round. After how many seconds will they meet at the starting point?

(a) 252 (b) 504 (c) 756 (d) 168

Answer: (b) 504

LCM(42, 56, 63):
  42 = 2 x 3 x 7
  56 = 2^3 x 7
  63 = 3^2 x 7
  LCM = 2^3 x 3^2 x 7 = 8 x 9 x 7 = 504

Q27. The smallest number which when divided by 5, 8, and 12 gives remainders 2, 5, and 9 respectively is:

(a) 113 (b) 117 (c) 120 (d) 237

Answer: (b) 117

Check deficit pattern:
  5 - 2 = 3
  8 - 5 = 3
  12 - 9 = 3
Deficit k = 3

LCM(5, 8, 12) = 120
Required = 120 - 3 = 117

Verification: 117/5 = 23 R 2; 117/8 = 14 R 5; 117/12 = 9 R 9. Correct.

Q28. 120 apples, 150 oranges, and 90 bananas are to be distributed equally among children. What is the maximum number of children?

(a) 10 (b) 15 (c) 30 (d) 60

Answer: (c) 30

HCF(120, 150, 90):
  120 = 2^3 x 3 x 5
  150 = 2 x 3 x 5^2
  90  = 2 x 3^2 x 5
  HCF = 2 x 3 x 5 = 30

Q29. Two numbers are in the ratio 3:4 and their HCF is 15. Their LCM is:

(a) 60 (b) 120 (c) 180 (d) 240

Answer: (c) 180

Numbers = 3 x 15 = 45 and 4 x 15 = 60
(Since ratio is 3:4 and HCF = 15, the numbers are 15 x 3 and 15 x 4)
Note: HCF(3, 4) = 1, confirming HCF of numbers = 15.

LCM = 15 x 3 x 4 = 180
Or: LCM = (45 x 60) / 15 = 180

Q30. The largest 4-digit number exactly divisible by 12, 15, 18, and 27 is:

(a) 9720 (b) 9828 (c) 9900 (d) 9960

Answer: (a) 9720

LCM(12, 15, 18, 27):
  12 = 2^2 x 3
  15 = 3 x 5
  18 = 2 x 3^2
  27 = 3^3
  LCM = 2^2 x 3^3 x 5 = 4 x 27 x 5 = 540

Largest 4-digit multiple of 540:
  9999 / 540 = 18.51...
  Floor = 18
  18 x 540 = 9720

Hard (Questions 31--42)

Q31. The HCF of two numbers is 12, their LCM is 720, and one number is 48. The other number is:

(a) 144 (b) 160 (c) 168 (d) 180

Answer: (d) 180

Other = (12 x 720) / 48 = 8640 / 48 = 180
Verification: HCF(48, 180) = 12; LCM = (48 x 180)/12 = 720. Correct.

Q32. Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10, and 12 seconds. In 30 minutes, how many times do they toll together?

(a) 15 (b) 16 (c) 14 (d) 10

Answer: (b) 16

LCM(2, 4, 6, 8, 10, 12):
  2  = 2
  4  = 2^2
  6  = 2 x 3
  8  = 2^3
  10 = 2 x 5
  12 = 2^2 x 3
  LCM = 2^3 x 3 x 5 = 120 seconds

In 30 minutes = 1800 seconds:
  Number of times they toll together = 1800/120 = 15

But they also toll together at time 0 (the start).
Total = 15 + 1 = 16

Q33. The LCM of two numbers is 45 times their HCF. If one number is 125 and the sum of HCF and LCM is 1150, find the other number.

(a) 200 (b) 225 (c) 250 (d) 275

Answer: (b) 225

LCM = 45 x HCF
HCF + LCM = 1150
h + 45h = 1150 --> 46h = 1150 --> h = 25
LCM = 45 x 25 = 1125

Other = (25 x 1125) / 125 = 28125 / 125 = 225

Verification: HCF(125, 225) = 25; LCM = (125 x 225)/25 = 1125; 1125/25 = 45. Correct.

Q34. Find the smallest number which when increased by 17 is exactly divisible by 520, 468, and 364.

(a) 32743 (b) 32760 (c) 32726 (d) 32777

Answer: (a) 32743

The number + 17 must be divisible by 520, 468, and 364.
So (number + 17) = LCM(520, 468, 364).

  520 = 2^3 x 5 x 13
  468 = 2^2 x 3^2 x 13
  364 = 2^2 x 7 x 13

  LCM = 2^3 x 3^2 x 5 x 7 x 13
      = 8 x 9 x 5 x 7 x 13
      = 32760

Number = 32760 - 17 = 32743

Verification:
  32743 + 17 = 32760
  32760 / 520 = 63 (exact)
  32760 / 468 = 70 (exact)
  32760 / 364 = 90 (exact)

Q35. Find the greatest number of 4 digits which is exactly divisible by 15, 24, and 36.

(a) 9720 (b) 9360 (c) 9000 (d) 9960

Answer: (a) 9720

LCM(15, 24, 36):
  15 = 3 x 5
  24 = 2^3 x 3
  36 = 2^2 x 3^2
  LCM = 2^3 x 3^2 x 5 = 360

Largest 4-digit multiple of 360:
  9999 / 360 = 27.77
  27 x 360 = 9720

Q36. The HCF and LCM of two numbers are 44 and 264 respectively. If the first number is divided by 2, the quotient is 44. The other number is:

(a) 132 (b) 128 (c) 148 (d) 160

Answer: (a) 132

First number / 2 = 44 --> First number = 88

Other = (HCF x LCM) / 88 = (44 x 264) / 88 = 11616 / 88 = 132

Verification: HCF(88, 132) = 44; LCM = (88 x 132)/44 = 264. Correct.

Q37. Three measuring rods are 64 cm, 80 cm, and 96 cm long. Find the least length of cloth that can be measured exact number of times using any of the rods.

(a) 720 cm (b) 960 cm (c) 1440 cm (d) 1920 cm

Answer: (b) 960 cm

LCM(64, 80, 96):
  64 = 2^6
  80 = 2^4 x 5
  96 = 2^5 x 3
  LCM = 2^6 x 3 x 5 = 64 x 15 = 960 cm

Q38. The ratio of two numbers is 4:5 and their HCF is 6. Their LCM is:

(a) 30 (b) 60 (c) 90 (d) 120

Answer: (d) 120

Numbers = 4 x 6 = 24 and 5 x 6 = 30
HCF(4, 5) = 1, confirming HCF of numbers = 6

LCM = 6 x 4 x 5 = 120
Or: (24 x 30) / 6 = 120

Q39. A, B, and C start running around a circular track at the same time from the same point. A completes a round in 252 seconds, B in 308 seconds, and C in 198 seconds. After what time will they all meet at the starting point?

(a) 2772 seconds (b) 5544 seconds (c) 13860 seconds (d) 27720 seconds

Answer: (a) 2772 seconds

LCM(252, 308, 198):
  252 = 2^2 x 3^2 x 7
  308 = 2^2 x 7 x 11
  198 = 2 x 3^2 x 11

  LCM = 2^2 x 3^2 x 7 x 11
      = 4 x 9 x 7 x 11
      = 2772

Verification: 2772/252 = 11, 2772/308 = 9, 2772/198 = 14. All exact.
2772 seconds = 46 minutes 12 seconds.

Q40. Find the largest number which divides 438 and 606 leaving remainders 6 and 12 respectively.

(a) 18 (b) 27 (c) 36 (d) 54

Answer: (d) 54

438 - 6 = 432
606 - 12 = 594

HCF(432, 594):
  594 = 432 x 1 + 162
  432 = 162 x 2 + 108
  162 = 108 x 1 + 54
  108 = 54 x 2 + 0

  HCF = 54

Verification: 438/54 = 8 R 6; 606/54 = 11 R 12. Correct.

Q41. The product of two numbers is 4107. If the HCF of these numbers is 37, the greater number is:

(a) 101 (b) 107 (c) 111 (d) 185

Answer: (c) 111

Let the numbers be 37x and 37y, HCF(x, y) = 1.
Product = 37x x 37y = 37^2 x xy = 4107
xy = 4107 / 1369 = 3

Co-prime pairs with product 3: (1, 3)
Numbers: 37 x 1 = 37 and 37 x 3 = 111

Greater number = 111

Q42. Three tankers contain 403 litres, 434 litres, and 465 litres of diesel. Find the maximum capacity of a container that can measure the diesel of each tanker exact number of times.

(a) 31 (b) 41 (c) 62 (d) 93

Answer: (a) 31

HCF(403, 434, 465):

  434 - 403 = 31
  465 - 434 = 31

Since differences are both 31, try HCF with 31:
  403 / 31 = 13 (exact)
  434 / 31 = 14 (exact)
  465 / 31 = 15 (exact)

  HCF = 31

Maximum capacity = 31 litres.

Answer Key

QAnswerQAnswerQAnswer
1(b) 1215(c) a divides b29(c) 180
2(b) 3616(c) 18030(a) 9720
3(b) 1217(c) 32231(d) 180
4(b) 218(a) 3532(b) 16
5(b) Co-prime19(c) 6533(b) 225
6(b) 2420(a) 7734(a) 32743
7(c) 12021(c) 250035(a) 9720
8(a) 2/1522(b) 236(a) 132
9(c) 7723(c) 3637(b) 960
10(d) Both b,c24(c) 7538(d) 120
11(c) 3625(a) 20/339(a) 2772
12(b) a divides b26(b) 50440(d) 54
13(c) 30027(b) 11741(c) 111
14(c) 628(c) 3042(a) 31

Next: 8.7 Quick Revision