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

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.

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) .

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.

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.

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.

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

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

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)

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

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

#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

(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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