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What is the number of positive integers less than 500 which

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What is the number of positive integers less than 500 which

by Max@Math Revolution » Tue Sep 03, 2019 11:30 pm

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[GMAT math practice question]

What is the number of positive integers less than 500 which have an odd number of divisors?

A. 18
B.20
C.22
D. 24
E. 26
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by deloitte247 » Sun Sep 08, 2019 10:44 am
The number of divisor of a number is the product of its exponents +1.
The square of an integer will always result in an odd number of divisors.
Therefore, the number of positive integers less than 500 which will have an odd number of the divisor is the square root of 500.
$$\sqrt{500}=22.36\approx22$$

Answer = C
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by Max@Math Revolution » Sun Sep 08, 2019 5:25 pm
=>

If the prime factorization of an integer n is n=(p^a)(q^b)(r^c), where p, q and r are primes, for example, then the number of factors of n is (a+1)(b+1)(c+1).
In order for (a+1)(b+1)(c+1) to be odd, a+1, b+1 and c+1 must all be odd, and a, b and c must be even numbers.
This means that a=2x, b=2y and c=2z for some integers x, y and z and n=(p^a)(q^b)(r^c)=(p^{2x})(q^{2y})(r^{2z})=((p^x)(q^y)(r^z))^2 must be the square of an integer.
The integer squares less than 500 are 1^2=1, 2^2=4, 3^2=9, ... , 22^2=484. There are 22 such squares.

Therefore, C is the answer
Answer: C
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