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How many numbers between 1 and 1000, inclusive have an odd n

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by Max@Math Revolution » Wed Apr 03, 2019 11:29 pm

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[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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by Scott@TargetTestPrep » Sun Apr 07, 2019 5:25 pm
Max@Math Revolution wrote:[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
Only perfect squares have an odd number of factors. Thus, the perfect squares between 1 and 1000, inclusive, are 1^2, 2^2, ..., 31^2, so there are a total of 31 numbers.

Answer: C

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by Max@Math Revolution » Sun Apr 07, 2019 5:34 pm
=>

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
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