Problem Solving

[GMAT math practice question]

How many numbers between 1 and 1000, inclusive have an odd number of factors?

A. 10
B. 25
C. 31
D. 64
E. 128

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Recall the property that an integer with an odd number of factors is the perfect square of an integer and only has even exponents in its prime factorization.
If n = p^aq^br^c, then n has (a+1)(b+1)(c+1) factors, where p, q and r are different prime numbers, and a, b and c are positive integers. So a + 1, b + 1, c + 1 must be odd numbers in order for n to have an odd number of factors. This implies that a, b and c are even numbers and n is the perfect square of an integer.
There are 31 perfect squares between 1 and 1000, inclusive, because 31^2 = 961 < 1000 and 32^2 = 1024 > 1000.

Therefore, the answer is C.
Answer: C