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If X is a positive integer, is X a perfect square?

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If X is a positive integer, is X a perfect square? [#permalink] New post   02 Jul 2022, 23:26
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67% (01:11) correct 33% (01:36) wrong based on 84 sessions
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If X is a positive integer, is X a perfect square?

(1) X has 10 factors
(2) The number of different factors of X is even


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Please try this question and let me know the explanation. My doubt in this question is the word 'factors' and not 'prime factors'. My thought is if X has 10 factors it can be 4, 10,13,...50 and not necessarily (prime)^9. For 2] it can be said that the product of (powers+1) of the prime factors is even meaning any of the (power+1) can be even. We cant conclude that all powers are even in that case. I would mark E then.
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Re: If X is a positive integer, is X a perfect square? [#permalink] New post   03 Jul 2022, 03:46
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Your thought process is correct.

However, you are making a mistake in the inference.

    When a number has 10 factors, you are correct to say that one of the power must be odd.
    Since one of the powers is odd, it cannot be a perfect square.
    Hence, this statement is sufficient and your answer is NO, it is not a perfect square.


You should also remember that all the perfect squares have an odd number of factors. And hence this number cannot be a prime number. So just looking at 10 factors, I can say the answer is NO. The statement is sufficient.
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Re: If X is a positive integer, is X a perfect square? [#permalink] New post   16 Nov 2023, 20:25
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1) X has 10 factors.

The only case X could have 10 factors if it is is p1p4, p4p1, or p9. Neither of these cases would give a perfect square so statement #1 is sufficient.

2) The number of different factors is even.

Take 2^2. The factors are 1, 2, 4. Therefore, a perfect square will always have an ODD number of different factors.

Therefore, D is the answer. Hope this helps.
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Re: If X is a positive integer, is X a perfect square? [#permalink] New post   16 Nov 2023, 22:06
Bottom line to note is that a prime number will have odd number of factors

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Re: If X is a positive integer, is X a perfect square? [#permalink]
16 Nov 2023, 22:06
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