If x and y are positive integer, what is the remainder when

That's a good question.
A. x=y(2k)+4, k any integer >=0.
B. x=y(p-1)+4, p any integer >=0.

Why A is not sufficient to determine the remainder and B is? Why did I use number plugging to show this in the first case and didn't in the second?

If we are told that x divided by y gives a remainder of 4, means x=yp+4 where p is integer >=0. We don't know x and y so p (coefficient) can be any integer.

Look at equation A, the coefficient is 2k, 2k is even. It can be rephrased as x divided by y will give the remainder of 4 IF coefficient is even. But what about the cases when coefficient is odd? We don't know that so we must check to determine this.

As for B. Coefficient here is (p-1), which for integer values of p can give us ANY value: any even as well as any odd. So basically x=y(p-1)+4 is the same as x=yp+4. No need for double checking.

Hope it's clear.

Thanks boss!!!

You will score +50 in your test definitely.

Asked: If x and y are positive integer, what is the remainder when x is divided by y ?

(1) When x is divided by 2x, the remainder is 4.
x = 4
Since y is unknown
NOT SUFFICIENT

(2) When x + y is divided by y, the remainder is 4.
Remainder when x is divided by y, the remainder will be 4.
SUFFICIENT

IMO B

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