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# If x ≠ 0, does x have an odd number of factors?

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Math Expert
Joined: 02 Sep 2009
Posts: 47983
If x ≠ 0, does x have an odd number of factors?  [#permalink]

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31 Jul 2018, 22:00
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62% (00:57) correct 38% (01:04) wrong based on 45 sessions

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If x ≠ 0, does x have an odd number of factors?

(1) √x is an integer.

(2) x^2 is an integer.

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Math Expert
Joined: 02 Aug 2009
Posts: 6554
Re: If x ≠ 0, does x have an odd number of factors?  [#permalink]

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31 Jul 2018, 22:28
If x ≠ 0, does x have an odd number of factors?

ODD number of factors :- Only perfect square have odd factors
so the question is :- Is x a perfect square?

(1) √x is an integer.
let √x=y, where y is an integer
so $$x=y^2$$, thus x is a perfect square
sufficient

(2) x^2 is an integer.
if x is an integer, ans is YES
but if x is not an integer, or example $$x^2=3......x=\sqrt{3}$$... ans is NO
insuff

A
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Re: If x ≠ 0, does x have an odd number of factors?  [#permalink]

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01 Aug 2018, 00:27
Hi,

This question, if you know below properties of perfect square numbers then its just a twenty second problem.

1). A perfect square will always have an odd number of factors

2). A square of a prime number will always have exactly 3 factors

3). A perfect square will have an odd number of odd factors and even number of even factors.

Let’s say if one student could not recollect this during the exam,then its always good to try some numbers.

Let’s say for an example,

If x = 2,3 then it has even number of factors. Infact all the prime numbers as even number of factors.

If x is an odd number (not prime), let’s say x = 15, then x has 1,3,5,15 four factors.

If x is an even number (not prime), let’s say x = 20, then x has 1,2,4,5,10 and 20 six factors.

Now lets x is 4 or 9, then the factors are 1,2, 4 and for 9 we have 1,3,9.

We can see that, perfect square numbers have odd number of factors.

This is infact because,

When we prime factorize as perfect square number, always the power of primes is even.

Then the number of factors would be odd.

For example, 36 = 2^2 * 3^2

So, the number of factors is equal to (2+1) * (2+1) = 9 factors.

During the exam, it won’t take more than 40 or 50 secs to check the possibilities(But I would still recommend to know the properties as these are commonly asked questions in GMAT).

Statement I is sufficient:

√x is an integer.

So, x is a perfect square number.

So sufficient.

Statement II is insufficient:

x^2 is an integer.

First thing is “x” here may or may not be an integer.

Let’s say x = √2 is not an integer. But x^2 is not an integer.

Even if we consider “x” is an integer, we are not still sure whether “x” is a perfect square number or not.

So not sufficient.

So the answer is A here.
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Re: If x ≠ 0, does x have an odd number of factors? &nbs [#permalink] 01 Aug 2018, 00:27
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