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

Question

If x and y are positive integers, what is the remainder when x is divided by y?

(1) When x is divided by 2y, the remainder is 4
(2) When x + y is divided by y, the remainder is 4

Bunuel · Accepted Answer

Positive integer  x  divided by positive integer  y  yields remainder of  r  can be expressed as  x=yq+r . Question is  r=?

(1) When x is divided by 2y, the remainder is 4. If  x=20  and  y=8  (satisfies the given statemnet as 20 divided by 2*8=16 yields reminder of 4), then  x  divided by  y  yields  r=4  (20 divided by 8 yields remainder of 4) BUT if  x=10  and  y=3  (satisfies the given statemnet as 10 divided by 2*3=6 yields reminder of 4), then  x  divided by  y  yields  r=1  (10 divided by 3 yields remainder of 1). Two different answers. Not sufficient.

(2) When x + y is divided by y, the remainder is 4 -->  x+y=yp+4  -->  x=y(p-1)+4  ( x  is 4 more than multiple of  y )--> this statement directly tells us that  x  divided by  y  yields remainder of 4. Sufficient.

Answer: B.

Hope it's clear.

KarishmaB · Answer

If you understand the concept of divisibility well, you can pretty much do this orally in less than 30 secs with a little bit of visualization. Divisibility involves grouping.

Stmnt 1: When x is divided by 2y, the remainder is 4

When you divide x by 2y, you make groups with 2y balls in each and you have 4 balls leftover. 
Instead, if you divide x by y, you may have 4 balls leftover or you may have fewer balls if y is less than or equal to 4 i.e. say if y = 3, you could make another group of 3 balls and you will have only 1 ball leftover. So you could have different remainders. Not sufficient.

Stmnt 2: When x + y is divided by y, the remainder is 4

When you make groups of y balls each from (x+y), the y balls make 1 group and you are left with x balls. If the remainder is 4, it means when you make groups of y balls each from x balls, you have 4 balls leftover.
Since the question asks us: how many balls are leftover when you make groups of y balls from x balls, you get your answer directly as '4'.
Sufficient.

Answer (B)

onedayill · Answer

1. IMO B

the second one: 
(x+y)/y= b+4, 
x/y+y/y=b+4, 
x/y+1=b+4 
or x/y=b+3. the remainder is 3
with this one we agree, it is sufficient.

bugSniper · Answer

I'm confused. With the same proof given for the second point, could I not convince myself that 1 would suffice?
eg: x = yp + r
From (1) I could say x = 2yp + 4
(or) x = (2p)y + 4
 Clearly this statement tells us that x is 4 more than a multiple of y as well. Why can I not convince myself at this step that the statement would suffice? Although, the statement does not suffice since the logic fails when you plug in (x=10,y=3) .

Bunuel · Answer

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 (quotient) can be any integer.

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

As for B. Quotient 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.

fluke · Answer

B is sufficient by using the rule of remainders additive property:
(x+y)/y leaves a remainder of 4.
Means: remainder left by x/y + remainder left by y/y = 4
remainder left by x/y+0=4
remainder left by x/y=4

At least B is Sufficient.

SonyGmat · Answer

Bunuel could we say the following? (Having in mind that the Remainder depends on the divisor)

(1) When x is divided by 2y, the remainder is 4

statement 1 ----> x=2*y*k+4, k integer

Therefore because the divisor has to be larger (not equal because it is stated that a reminder exists) than the remainder: 2*y>4 --> y>2 -->y>=3

If y (divisor) is smaller than 4 then the remainder changes and if it is larger than 4 the remainder is 4.

For example:
if y=3 then x=2*3*k+3+1, R=1
if y=4 then x=4*2*k+4+0, R=0
(if y=5 then x=2*5k+4, R=4)

Therefore Insufficient.

2) When x + y is divided by y, the remainder is 4

statement 2 ----> x+y=y*k+4, k integer

We are told that the remainder is 4, therefore y>=5! Which means that remainder will always be 4.

manulath · Answer

Statement 1 is insuff.
x=4/3, y=3/2, x=8, y=9.......

Statement 2:
(x+y)/y has remainder as 4
this can be written as.......(x/y + 1)
as 1 is a whole number the fraction comes from x/y.......which is 4

Hence SUFF.

Ans B

jlgdr · Answer

Answer is (B) this is pretty straightforward. Statement 1: x= 2y+4 and we also know that we want to find x=4+r so if we equal both we get y+4 = r. We don't know 'y' so not good enough Statement 2: x+y/y remainder is 4. y/y has no remainder so x/y will have remainder 4. Hence it is (B) Hope it helps Kudos if you like

KarishmaB · Answer

Responding to a pm:
Before I analyze this question for you, you must understand the method I use for divisibility (making groups).

Statement 1 alone is not sufficient. Here is the reason:
1) When x is divided by 2y, the remainder is 4
This means that when you split x balls into groups of 2y balls each, you are left with 4 balls out of which you cannot make any more groups. (So 2y must be greater than 4 and y must be greater than 2)
What if instead, you had to split those balls into groups of y balls each. You made groups of 2y balls each above. Now you can easily split each oen of those groups into two groups of y balls each. But what about the 4 leftover balls? Can you make another group of x balls out of it? Depends on the value of x. Say if x is 3, you can make another group and the remainder will be 1. If x is 5, you cannot make another group and remainder will still be 4. Hence this statement is not sufficient to say what the remainder is when x is divided by 2y.

Statement 2 alone is sufficient.
2) When x + y is divided by y, the remainder is 4
You have x+y balls and you have to make groups out of them containing y balls each. The y balls make one group and the x balls make more groups with 4 balls remaining. Hence when you make groups of y balls each out of x balls, you get a remainder of 4. This means that when you divide x by y, you get remainder 4.

Answer (B)

CrackverbalGMAT · Answer

Statement I is insufficient:

x = 10, y = 3
x = 20, y = 8

Statement II is sufficient:

(x + y)/y = (x/y) + 1 the remainder is 4 hence we know the remainder when x is divided by y

Answer is B

EgmatQuantExpert · Answer

Here's an alternate solution to this question, without using number-plugging: The question asks us the remainder for x/y This means, in the expression x = ky + r . . . (1) (where k is the quotient and r is the remainder) We have to find the value of r, where 0 =< r < y Analyzing St. 1 When x is divided by 2y, the remainder is 4 That is, x = (m)(2y) + 4 . . . (2) Now, upon comparing Equations (1) and (2), can we say r = 4? Let's see: We know that remainder is always less than the divisor. Since r is the remainder in Equation 1, we can be sure that r < y Similarly, since 4 is the remainder in Equation 2, we can be sure that 4 < 2y. This gives us, y > 2 So, y can be 3, 4, 5, 6, 7 etc. Now, if y = 3, then r must be less than 3 . So, upon comparing (1) and (2), you'll not say that r = 4. Instead, you'll break down 4 into 3 + 1. So, if y = 3, remainder r = 1 If y = 4, then r must be less than 4 . So, upon comparing (1) and (2), you'll not say that r = 4. Instead, you'll write 4 as 4 + 0. So, if y = 4, remainder r = 0 If y > 4, then 4 becomes a valid value for remainder r . So, in this case, upon comparing (1) and (2), we can say that r = 4. Thus, we see that r can be 0, 1 or 4. Since we have not been able to determine a unique value of r, St. 1 is not sufficient. Analyzing St. 2 When x + y is divided by y, the remainder is 4 That is, x + y = qy + 4 Or, x = (q - 1)y + 4 . . . (3) Equation (3) conveys that when x is divided by y, the remainder is 4. So, St. 2 is sufficient to find the value of the remainder. Please note that the way we processed St. 1 was different from the way we processed St. 2, because in St. 1 the divisor was 2y, whereas in St. 2, the divisor was y (the same divisor as in the question) Hope this alternate solution was useful!

Best Regards Japinder

Archit3110 · Answer

#1 When x is divided by 2y, the remainder is 4 let x = 4 and y can be 3,4,5,6,7,8 we observe that remainder of x/y ; varies from 0,1,4,1/2 so insufficient #2 When x + y is divided by y, the remainder is 4 or say x/y+y/y = 4 remainder of y/y will be 0 therefore x/y remainder is 4 sufficient OPTION B

smmc29 · Answer

(1) x/2y remainder is 4

Take 2 representative values for y. y should be greater than 2 since the denominator should be greater than 4 to yield a remainder equal to 4.

When y = 3

x/(2x3) then x should be 10,16,22,... When x is divided by y, the remainder will always be 1

When y=5
x/(2x5) then x should be 14,24,... the remainder will always be equal to 4

INSUFFICIENT

(2) (x+y)/y remainder is 4

Simplify the expression we have x/y + 1.

This simply indicates that the remainder of x/y is 4.

SUFFICIENT

The answer is B.

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

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