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23 Oct 2024, 01:30
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
35%
(medium)
Question Stats:
73% (01:51) correct
27%
(02:24)
wrong
based on 218
sessions
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Date
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Not Attempted Yet
If \(375y = x^2\), and \(x\) and \(y\) are positive integers then which of the following must be an integer?
I. \(\frac{y}{15}\)
II. \(\frac{y}{30}\)
III. \(\frac{(y^2)}{25}\)
A. I only
B. III only
C. I and II
D. I and III
E. I, II and III
I. \(\frac{y}{15}\)
II. \(\frac{y}{30}\)
III. \(\frac{(y^2)}{25}\)
A. I only
B. III only
C. I and II
D. I and III
E. I, II and III
This is a PS Butler Question
Check the links to other Butler Projects:
Check the links to other Butler Projects:
- Data Sufficiency Butler
- Critical Reasoning Butler
- Reading Comprehension Butler
- Integrated Reasoning Butler
23 Oct 2024, 08:49
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We realize that \(375y\) is a perfect square, because x is an integer.
Additionally, we should know that a “Perfect square always has even number of powers of prime factors” (GMAT Club Math Book v8 2024, p. 7).
So let's do prime factorization: \(375=5*75=5*25*3=5*5*5*3=3^1*5^3 \)
We can see that the powers of 5 and 3 are both not even.
So to get them even there has to be AT LEAST one 5 and one 3 in the prime factorization of y:
\(y=3*5*n = 15n\) , where n is a positive integer
I. \(\frac{y}{15}= \frac{15n}{15} = n \) => CORRECT
II. \(\frac{y}{30} = \frac{15n}{30} = \frac{n}{2} \) => if n=1 than \(\frac{1}{2}\) => INCORRECT
III. \(\frac{y^2}{25} = \frac{(15n)^2}{25} = \frac{3^25^2n^2}{5^2} = 3^2*n^2 \) => CORRECT
ANSWER: D
Additionally, we should know that a “Perfect square always has even number of powers of prime factors” (GMAT Club Math Book v8 2024, p. 7).
So let's do prime factorization: \(375=5*75=5*25*3=5*5*5*3=3^1*5^3 \)
We can see that the powers of 5 and 3 are both not even.
So to get them even there has to be AT LEAST one 5 and one 3 in the prime factorization of y:
\(y=3*5*n = 15n\) , where n is a positive integer
I. \(\frac{y}{15}= \frac{15n}{15} = n \) => CORRECT
II. \(\frac{y}{30} = \frac{15n}{30} = \frac{n}{2} \) => if n=1 than \(\frac{1}{2}\) => INCORRECT
III. \(\frac{y^2}{25} = \frac{(15n)^2}{25} = \frac{3^25^2n^2}{5^2} = 3^2*n^2 \) => CORRECT
ANSWER: D
General Discussion
23 Oct 2024, 02:57
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It is given that 375y=x^2,
5×5×5×3×y=x^2,
✓5×5×5×3×y=x,
5✓15y=x.
Now it is already given that x and y both are +ve integers,so
5✓15y>0,
✓15y>0 (dividing both sides by 5)
Now for ✓15y to be greater than 0 y must be ✓15 which is ✓5×3,so to remove bar,we need one 5 and one 3,which means y should be 15 or a number divided by 15.
Applying this on the options will give us our answer.
😊
5×5×5×3×y=x^2,
✓5×5×5×3×y=x,
5✓15y=x.
Now it is already given that x and y both are +ve integers,so
5✓15y>0,
✓15y>0 (dividing both sides by 5)
Now for ✓15y to be greater than 0 y must be ✓15 which is ✓5×3,so to remove bar,we need one 5 and one 3,which means y should be 15 or a number divided by 15.
Applying this on the options will give us our answer.
😊
