Quotients

Question

I got the right answer but I don't really know how to solve it correctly, any Help would be very valuable thanks!

When the positive integer x is divided by 11, the quotient is y and the remainder 3. When x is divided by 19, the remainder is also 3. What is the remainder when y is divided by 19?

A) 0
B) 1
C) 2
D) 3
E) 4

IanStewart · Answer

I can suggest three ways to look at this: by the quotient/remainder formula, when we divide n by d, we have that n = qd + r, where q is the quotient, and r is the remainder; here it must be that 0  <  r < d.

So from the information in the question, we know that x = 11y + 3, and also that x = 19q + 3, for some integer quotient q. If x is equal to both of those expressions, those two expressions are equal, so we have

11y + 3 = 19q + 3
11y = 19q

Now both sides of that equation are integers; the primes which divide the right side must divide the left. So 19 must be a factor of 11y, and since it is not a divisor of 11, it must be a divisor of y. Thus the remainder is zero when y is divided by 19.

Or, perhaps a bit more abstract: If the remainder is 3 when x is divided by 11, then x - 3 is a multiple of 11. Similarly, from the information given, x - 3 is a multiple of 19. So x - 3 = 11*19*k, for some integer k, and x = 11*19*k + 3. Since y is the quotient when x is divided by 11, from this last equation y must be equal to 19k, so y is a multiple of 19, and the remainder is zero when y is divided by 19.

Finally, if you can find a number x might equal here, you can solve the problem with that number. Since there is no answer choice which reads 'cannot be determined', if we can solve the problem for one value of x, we can be confident the answer is always the same for *any* value of x, since otherwise the question would not have only one correct answer -- and if it's a GMAT question, it can only have one correct answer. If the remainder is 3 when x is divided by both 11 and 19, then the simplest value we could choose for x is 3 itself. Then the quotient y is 0 when x is divided by 11, so the remainder will be 0 when y is divided by 19.

deepakraam · Answer

The question can be re-framed as ; so x = 11y + 3

x / 19 can also be written as (11y+3)/19

The remainder of the above operation can be got : Rem (11y/19) + Rem (3/19)

Rem( 11y/19) + 3 = 3

So,Rem (11y/19) = 0

I wud go with option A)0

PussInBoots · Answer

Plugging in numbers would be the best solution

x could be 11 * 19 + 3, hence y = 19

gmate2010 · Answer

Nothing can be better than this..

..very impressive..Thanks

srivas · Answer

I got the right answer but I don't really know how to solve it correctly, any Help would be very valuable thanks!

When the positive integer x is divided by 11, the quotient is y and the remainder 3. When x is divided by 19, the remainder is also 3. What is the remainder when y is divided by 19?

A) 0
B) 1
C) 2
D) 3
E) 4

Soln: Since the number when divided by 11 or 19 gives the same remainder, therefore
x = LCM(11,19)* k + 3 where k = 0,1,2,3
x = 209k + 3 taking k = 1
x = 209 * 1 + 3
x = 212

Now since its given that the quotient when x is divided by 11 is y, we have
x = 11 * y + 3
202 = = 11 * y + 3
from this we can solve for y = 19

hence when y is divided by 19 we have remainder as 0

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