Nevernevergiveup wrote:

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

Given, \(375y = x^2\) ---> as x and y are both integers ---> 375*y must become a perfect square for x to be an integer

375*y ---> \(3*5^3*y\)

Thus, from the given information, \(x^2=3*5^3*y\) ---> y MUST be = \(p*3^{odd}*5^{odd}\), where p = integer. Possible values of y = \(3*5\) or \(3^3*5\) or \(3^3*5^3\) etc and in all cases you can see that y = 15p or y is a multiple of p.

Going back to the options,

I) y/15 MUST be an integer as shown above.

Must be a true statement.II) y/30, as we see that y = 15p but it may or may not be even multiple of 15. y = 15 or 75 are equally applicable to the conditions given but is not divisible by 30. Hence this statement is

not a must be true statement.

III) y^2/25 is an integer . As y = 15p ---> y^2 = 225p = 25*9p = 25*q. Thus y^2 is a multiple of 25 and hence this

statement is a must be true.

D is thus the correct answer.