7TH NUMBERS TALENT- HCF -WORD PROBLEMS- 6

Highest Common Factor (HCF) Aptitude Quiz

1. Find the Highest Common Factor (HCF) of 144 and 180.

A) 12 B) 18 C) 36 D) 72

Correct Answer: C) 36

Explanation:
144 = 2⁴ × 3²
180 = 2² × 3² × 5
HCF = 2² × 3² = 4 × 9 = 36.

2. What is the HCF of 42, 63, and 105?

A) 7 B) 14 C) 21 D) 42

Correct Answer: C) 21

Explanation:
42 = 2 × 3 × 7
63 = 3² × 7
105 = 3 × 5 × 7
HCF = 3 × 7 = 21.

3. What is the HCF of two consecutive even numbers?

A) 1 B) 2 C) 4 D) Depends on the numbers

Correct Answer: B) 2

Explanation: Any two consecutive even numbers can be represented as 2n and 2n + 2. The greatest common factor shared by both numbers is always 2. Example: HCF(10, 12) = 2.

4. Calculate the HCF of 108, 288, and 360.

A) 18 B) 24 C) 36 D) 54

Correct Answer: C) 36

Explanation:
108 = 2² × 3³
288 = 2⁵ × 3²
360 = 2³ × 3² × 5
HCF = 2² × 3² = 4 × 9 = 36.

5. Find the HCF of the fractions 2/3, 8/9, and 16/81.

A) 2/81 B) 16/3 C) 2/3 D) 8/81

Correct Answer: A) 2/81

Explanation: HCF of fractions = HCF of Numerators / LCM of Denominators.
HCF(2, 8, 16) = 2
LCM(3, 9, 81) = 81
Result = 2/81.

6. A sweet seller has 420 Kaju Barfis and 130 Badam Barfis. She wants to stack them in equal rows. What is the maximum number of barfis per stack?

A) 10 B) 20 C) 30 D) 15

Correct Answer: A) 10

Explanation: To find the maximum number per stack, calculate HCF(420, 130).
420 = 130 × 3 + 30
130 = 30 × 4 + 10
30 = 10 × 3 + 0
HCF is 10.

7. Three ropes of length 48 m, 80 m, and 96 m are to be cut into equal pieces of maximum length. What is the length of each piece?

A) 8 m B) 12 m C) 16 m D) 24 m

Correct Answer: C) 16 m

Explanation: Maximum equal piece length = HCF(48, 80, 96).
48 = 2⁴ × 3
80 = 2⁴ × 5
96 = 2⁵ × 3
HCF = 2⁴ = 16 m.

8. A milkman has 45L cow milk, 60L toned milk, and 75L full-cream milk. What is the maximum capacity container to measure all three without mixing?

A) 5 liters B) 15 liters C) 25 liters D) 30 liters

Correct Answer: B) 15 liters

Explanation: Required capacity = HCF(45, 60, 75).
45 = 3² × 5
60 = 2² × 3 × 5
75 = 3 × 5²
HCF = 3 × 5 = 15 liters.

9. Find the greatest number that will divide 137, 182, and 422 leaving a remainder of 2 in each case.

A) 15 B) 30 C) 45 D) 60

Correct Answer: C) 45

Explanation: Subtract remainder 2 from each given number:
137 - 2 = 135
182 - 2 = 180
422 - 2 = 420
Calculate HCF(135, 180, 420) = 45.

10. A rectangular courtyard 18 m 72 cm long and 13 m 20 cm broad is paved with square tiles. What is the side length of the largest square tile?

A) 24 cm B) 36 cm C) 42 cm D) 12 cm

Correct Answer: A) 24 cm

Explanation: Convert to centimeters:
Length = 1872 cm, Breadth = 1320 cm.
Largest square tile side = HCF(1872, 1320) = 24 cm.

11. What is the greatest number that divides 260, 1314, and 1331 leaving remainders of 5, 6, and 11 respectively?

A) 15 B) 17 C) 19 D) 23

Correct Answer: B) 17

Explanation: Subtract individual remainders:
260 - 5 = 255
1314 - 6 = 1308
1331 - 11 = 1320
HCF(255, 1308, 1320) = 17.

12. The product of two numbers is 2028 and their HCF is 13. How many such possible pairs of numbers can be formed?

A) 1 B) 2 C) 3 D) 4

Correct Answer: B) 2

Explanation: Let numbers be 13a and 13b (where a, b are co-prime).
13a × 13b = 2028 → 169ab = 2028 → ab = 12.
Co-prime pairs for ab = 12 are (1, 12) and (3, 4). Total pairs = 2.

13. The ratio of two numbers is 3:4. If their HCF is 15, what is the sum of the two numbers?

A) 105 B) 120 C) 135 D) 150

Correct Answer: A) 105

Explanation: Let numbers be 3x and 4x. Here x is the HCF, so x = 15.
First number = 3 × 15 = 45
Second number = 4 × 15 = 60
Sum = 45 + 60 = 105.

14. What is the HCF of (2³ × 3² × 5), (2² × 3³ × 5²), and (2⁴ × 3 × 5³ × 7)?

A) 30 B) 60 C) 120 D) 180

Correct Answer: B) 60

Explanation: Take lowest power of common prime factors:
Lowest power of 2 is 2².
Lowest power of 3 is 3¹.
Lowest power of 5 is 5¹.
HCF = 2² × 3 × 5 = 4 × 3 × 5 = 60.

15. Find the greatest number which divides 62, 132, and 237 leaving the same remainder in each case.

A) 25 B) 35 C) 45 D) 55

Correct Answer: B) 35

Explanation: Find HCF of absolute differences:
|132 - 62| = 70
|237 - 132| = 105
|237 - 62| = 175
HCF(70, 105, 175) = 35.