Single correct option
How many positive divisors of 1080 are multiples of 6?
- A. 12
- B. 16
- C. 18
- D. 24
Show answer and solution
Answer: (C) 18
1080 = 2³ × 3³ × 5.
A divisor 2ᵃ × 3ᵇ × 5ᶜ is a multiple of 6 when a ≥ 1 and b ≥ 1.
Choices: a ∈ {1, 2, 3} (3 ways), b ∈ {1, 2, 3} (3 ways), c ∈ {0, 1} (2 ways).
Count = 3 × 3 × 2 = 18.
Alternative: Such divisors are 6k where k divides 1080/6 = 180 = 2² × 3² × 5, which has 3 × 3 × 2 = 18 divisors.
A divisor 2ᵃ × 3ᵇ × 5ᶜ is a multiple of 6 when a ≥ 1 and b ≥ 1.
Choices: a ∈ {1, 2, 3} (3 ways), b ∈ {1, 2, 3} (3 ways), c ∈ {0, 1} (2 ways).
Count = 3 × 3 × 2 = 18.
Alternative: Such divisors are 6k where k divides 1080/6 = 180 = 2² × 3² × 5, which has 3 × 3 × 2 = 18 divisors.
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