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

Question

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

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.

SaquibHGMATWhiz · Answer

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.

juliefengmba · Answer

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.

Maheshkv · Answer

Bottom line to note is that a prime number will have odd number of factors

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architkap · Answer

could someone please explain statement 1 and how its sufficient?

Bunuel · Answer

If X is a positive integer, is X a perfect square?  
 
A positive integer is a perfect square  if and only if  it has an  odd number of positive factors .

(1) X has 10 factors.

10 is even, so X cannot be a perfect square. Sufficient.

(2) The number of different factors of X is even.

A perfect square must have an odd number of different positive factors. Therefore, X is not a perfect square. Sufficient.

Answer: D.

Pure algebraic questions are no longer a part of the  DS syllabus  of the GMAT.

DS questions in GMAT Focus encompass various types of word problems, such as:

Word Problems 
 Work Problems 
 Distance Problems 
 Mixture Problems 
 Percent and Interest Problems 
 Overlapping Sets Problems 
 Statistics Problems 
 Combination and Probability Problems

While these questions may involve or necessitate knowledge of algebra, arithmetic, inequalities, etc., they will always be presented in the form of word problems. You won’t encounter pure "algebra" questions like, "Is x > y?" or "A positive integer n has two prime factors..."

Check  GMAT Syllabus for Focus Edition

You can also visit the  Data Sufficiency forum  and filter questions by  OG 2024-2025, GMAT Prep (Focus), and Data Insights Review 2024-2025 sources  to see the types of questions currently tested on the GMAT.

So, you can ignore this question.

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

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GMAT Data Sufficiency (DS) Questions

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

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