How many positive divisors of 7,200 are perfect squares?
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Answer & Explanation
Correct answer
D
Factor 7,200 = 72 × 100 = (2³ × 3²) × (2² × 5²) = 2⁵ × 3² × 5². A divisor is a perfect square exactly when every prime's exponent in it is even, so count the even-exponent options for each prime independently.
For 2⁵ the even exponents are 0, 2, 4, giving 3 options; for 3² they are 0, 2, giving 2 options; for 5² they are 0, 2, giving 2 options. The number of perfect-square divisors is 3 × 2 × 2 = 12.
As a cross-check, enumerating the square divisors directly gives 1, 4, 9, 16, 25, 36, 100, 144, 225, 400, 900, 3600, which is 12 values. The total divisor count (5+1)(2+1)(2+1) = 54 is choice E, which ignores the even-exponent restriction.