If y and z are factors of x^2, is x divisible by y? (1) y is a prim

would be interesting to see the responses to this question

1. SUFFICIENT: Any prime number that is a factor of x^2 must also be a factor of x. If x=y*z, x^2 = (y^2)*(z^2) OR (yz)^2.
2. SUFFICIENT: z = y*y. x is divisible by z, therefor x is divisible by (y*y).

Answer D, both sufficient separately.

Question is asking if \(\frac{x}{y}\) is an integer.

We could pick numbers

S1)
\(x=2\). Hence \(x^{2}=4\). Factors of \(x^{2}\) are 1, 2, and 4.
Since S1 says that \(y\) is prime, \(y\) could only be 2.
Hence \(\frac{x}{y}\)=\(\frac{2}{2}=integer\) YES.

Lets pick another number to see if we can get a NO.

\(x=-6.\) Hence \(x^{2}=36\). Factors of \(x^{2}\) are 1,2,3,4,6,9,12,18,and 36.
Since S1 says that \(y\) is prime, \(y\) could now be 2 or 3 only.
Hence in both the cases \(\frac{x}{y}=integer\). YES.

So we have tried a prime number and a composite number for \(x\) and in both the cases the answer is YES. So Statement 1 is sufficient. Answer Choices could be A or D.

S2) Since we are at A or D, we could verify if statement 2 is definitely true using information from Statement 1, and if proved answer would be D.(Because Statement 1 returned a definite YES and it cannot be contradicted)

Statement 2 says \(z=y^{2}\).

If we look into the first plug-in that we did in S1, \(z\) could only be 4 meeting the condition of S1 that \(y\)=Prime=2. There was no other possible value. And it can be seen that \(z =4\) is a factor of \(x^{2}= 4\) meeting the condition in question stem also. So statement 2 is proved using statement 1.

We could try for the second plug-in we did in statement 1 also. Here \(z\) could only be 4 or 9 meeting the condition of S1 that\(y=prime=2\) or \(3\). There was no other possible value. And it can be seen that \(z=4\) or \(9\) is a factor of \(x^{2}=36\), meeting the condition in question stem also. So statement 2 is proved using statement 1.

Hence answer is D.

Hi Harsha

What if the value of x is sqt(2)? i.e x is not integer.
Would it still be true for statement 1?

x=sqt(2)--> x^2=2--> Factore 1,2
y could be 2 as y is prime and z could be 1
but in this case x wouldn't be divisible by y.
hence, statement one is not suffice.

Please correct me if I'm missing something.

We've taken the assumption that x is an integer, and in this case both statements are individually sufficient.

But the problem is that the question states that x^2 is an integer, not x. What if x is an irrational number?

Let's assume \(x = \sqrt{18}.\)

(1) y is a prime number.

y = 3 is a factor of x^2 = 18 but not a factor of x, because it's not an integer. Insufficient.

(2) z = y^2

z = 9 which is a factor of x^2 but again not a factor of x. Insufficient.


Bot statements taken together are giving y=3 and z=9 are factors of x^2 but not factors of x. Again insufficient.

Answer should be E.

you cant assume x is an integer unless mentioned in the question stem. Therefore I believe the answer should be E.
Experts, please comment.

Agreed totally. I also believe the same.

Expert's responses will further clarify if I am really missing something.

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The question is carelessly worded, and if this were a real GMAT question, it would always tell you that x is a positive integer.

As the question is written, it allows the possibility that x is some square root of an integer, say √2 or √3. But the concept of "divisibility" that is tested on the GMAT only makes sense when you're discussing integers. It's only in math far beyond GMAT-level that you'd learn what a question like "is √2 divisible by 2?" even means. Using GMAT concepts only, the question is like asking "is √2 orange?" The GMAT can't ask you questions that don't make sense, so the GMAT will always rule out the possibility that something is a non-integer in a divisibility problem.

If the question states that x is a positive integer, then it's a more realistic question, and the answer is D.
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I agree as well. It does not mention that x is an integer so x can very well be a non-integer.


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Note- % represents remainder
Given-
\(\frac{x^2}{y }\) % 0
\(\frac{x^2}{z }\) % 0

To find-
\(\frac{x}{y }\) % 0

1. y is a prime number
Since y is a prime number, a multiple of y must be present in x^2 and therefore even if we take root of x^2 the factor of y will still be present in x
Consider, y=p
x^2 = a^2 p^2
then, x = ap

2. z=y^2
\(\frac{x^2}{z }\)% 0 = \(\frac{x^2}{y^2 }\)% 0
Also we know, \(\frac{x^2}{y }\)% 0
Now since x^2 is divisible by both y and y^2. We understand that x^2 has the multiple of y^2 or maybe y^2 itself and if we take square root of x^2, it will still be divisible by y
Consider, x^2 = a^2 b^2 so x = ab
y^2 = b^2 and y = b

Correct ans D