When the integer x is divided by the integer y, the remainder is 60
=> x = y*q + 60, where q is the quotient

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Theory
Now the question is interesting as they have written the quotient as a decimal number. Now, lets understand that using a simple example first
20 when divided by 4 gives 5 as quotient, that's straight forward right.
20 when divided by 3 gives 6 as quotient and 2 as remainder, that's easy too
Now, 20 when divided by 3 gives 6.67 (approx.) as quotient. Now this is what they mean when they say that x/y quotient is 15.15 in option 1.
Lets go bit deeper into this, this means that 20 when divided by 3 gives 0.67 * 3 as remainder, which is nothing but 2
So, 0.67*3 will be the remainder
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Now, with that context lets attempt the problem.

I. \(\frac{x}{y}\) quotient is 15.15 => 0.15*y = 60 => y = \(\frac{60}{0.15}\) => y = 400 => Possible as y is integer
II. \(\frac{x}{y}\) quotient is 18.16 => 0.16*y = 60 => y = \(\frac{60}{0.16}\) => y = 375 => Possible as y is integer
III. \(\frac{x}{y}\) quotient is 17.17 => 0.17*y = 60 => y = \(\frac{60}{0.17}\) => y = 352.3 => NOT Possible as y is NOT an integer

So, I and II are correct
So, Answer will be D
Hope it helps!

1) GRE Lesson: Remainders Basics (PSAT\SAT\ACT\GMAT\GRE\Bank PO)



2) GRE Lesson: Remainders Advanced



You can read more about Remainders in this post.