Perfect Squares

Question

Hi, I'm struggling to understand the concept behind a particular perfect squares question. Could anybody help?

If 375y = x^2, and x and y are positive integers then which of the following must be an integer?

I. y/15
II. y/30
III. y^2/25

As I'm new to this forum, please do advise if I've used incorrect symbols or haven't laid everything out as per standard practice. Thanks in advance.

bb · Answer

Welcome to GMAT Club!
There are easier ways to write the math symbols here - I have edited your expression below. You can learn how to post math symbols here: writing-mathematical-symbols-in-posts-72468.html. Usually, all you have to do is to highlight the text and hit the M button. Bu there are exceptions such as fractions.

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. \(\frac{y^2}{25}\)

Let me add another twist - a Stopwatch

tkarthi4u · Answer

if 375 * y = x ^2

x and y are integers u can write it as (5^3 * 3) * y = x^2

since x is an integer x ^2 is a perfect square . so to make LHS a perfect square y = 5 *3

so u can see y/15 and y^2/25 is an integer.

i would go for i and iii.

joyseychow · Answer

Welcome on board!

1. Looking 375y=x^2 tells us that y has to also be 375 in order for x to be a perfect square. 2. Breakdown 375 to its prime factor---- (5*5*5*3)=y 3. Now it's much easier to compare each statement. I. y/15----> Yes (quotient=25) II.y/30----> No (quotient=25/2) III.y^2/25----> Yes (quotient=5625)

Jivana · Answer

Just as 3121gmat, I'm having some difficulty in understanding this problem.

Here's my 2 cents:

y doesn't have to be 375. For example, if we had a statment like 9y = x^2, which will be valid if y=4 & x=6.

In this case, if y=15, x^2 = 375*15,
or x=75.

I will go with (I)

sudeep · Answer

Let me give my shot

375y=x^2 
splitting the 375 =  5^2*5*3 
now as 375y is a square of positive integer and y is also a positive integer
implies that y has to be = 5*3*a^2, where a is an integer too

Why? let me show
375y= 5^2*5*3*   =  25^2*3^2*a^2

a^2  is added so as to generale the possible values of y. 
If you don't understand till here try different values of 'a' such as {2,3,4} to get y and x. It will give you different set of values to understand the reason of using a.

Now start with problem

1)  y/15 = 15*a^2/15 = a^2 
definitely a integer that too positive

2)  y/30 = 15*a^2/30 = a^2/2 
not necessary an integer if a^2 is not a multiple of 2.

3)  y^2/25 = 15*15*a^4/25 = 9a^4 
definitely an integer.

(i) and (iii)

lexis · Answer

375y= 3.5.5.5.y = x^2 ==> the number of 3 is even number, the number of 5 is even number ==> y contains both 3 and 5.

Clearly, (I) and (III) meet requirement.

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