Single correct option
How many divisors of 1800 are perfect squares?
- A. 6
- B. 8
- C. 9
- D. 12
Show answer and solution
Answer: (B) 8
1800 = 2³ × 3² × 5².
A square divisor must have even exponents: power of 2 ∈ {0, 2} (2 choices), power of 3 ∈ {0, 2} (2 choices), power of 5 ∈ {0, 2} (2 choices).
Number of square divisors = 2 × 2 × 2 = 8 (1, 4, 9, 25, 36, 100, 225, 900).
Tip: For exponent a, the number of even choices is ⌊a/2⌋ + 1.
A square divisor must have even exponents: power of 2 ∈ {0, 2} (2 choices), power of 3 ∈ {0, 2} (2 choices), power of 5 ∈ {0, 2} (2 choices).
Number of square divisors = 2 × 2 × 2 = 8 (1, 4, 9, 25, 36, 100, 225, 900).
Tip: For exponent a, the number of even choices is ⌊a/2⌋ + 1.
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