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# When the positive integer x is divided by 11, the quotient

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When the positive integer x is divided by 11, the quotient [#permalink]

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21 Jan 2012, 01:21
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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

Guys struggling to solve this. But this is the concept I am trying to apply:

We can extrapolate a general statement from this form. When dividing x by y, the quotient is q and the remainder is r:
x/y = q + r/y

From there, we can solve for x:
x = qy + r (thats the general form of x = 3(integer) + 1)
Or the quotient:
q = x-r/y

Or, even, the remainder itself:
r = x - qy

But I am getting stuck in finding y when x is divided by 19. Can someone please help?? I don't have an OA either.
[Reveal] Spoiler: OA

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21 Jan 2012, 03:02
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enigma123 wrote:
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

If you decide to go with quotient/remainder formula approach, then I'd suggest to express the info in the stem with it. And then look whether we can somehow manipulate with the expressions at hand to answer the question.

(1) When the positive integer x is divided by 11, the quotient is y and the remainder 3 --> $$x=11y+3$$;
(2) When x is divided by 19, the remainder is also 3 --> $$x=19q+3$$.

Easy to spot that $$19q+3=11y+3$$ --> $$19q=11y$$ --> $$y=\frac{19q}{11}$$. Now as $$y$$ and $$q$$ are integers and 19 is prime then $$\frac{q}{11}$$ must be an integer --> $$y=19*integer$$ --> $$y$$ is a multiple of 19, hence when divide by 19 remainder is 0.

Hope its clear.
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Re: When the positive integer x is divided by 11, the quotient [#permalink]

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21 Jan 2012, 03:13
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Find some numbers that fit the question.

We are told that x and y both give a remainder of 3 when divided by 19 or 11.

We can easily construct a number that fits this.

x = 19*11 + 3

x / 11 = 19 + 3 / 19

so y = 19

remainder when y / 19 is 0

A

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Re: When the positive integer x is divided by 11, the quotient [#permalink]

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12 Dec 2013, 05:40
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Re: When the positive integer x is divided by 11, the quotient [#permalink]

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10 Mar 2014, 21:54
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enigma123 wrote:
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

Guys struggling to solve this. But this is the concept I am trying to apply:

We can extrapolate a general statement from this form. When dividing x by y, the quotient is q and the remainder is r:
x/y = q + r/y

From there, we can solve for x:
x = qy + r (thats the general form of x = 3(integer) + 1)
Or the quotient:
q = x-r/y

Or, even, the remainder itself:
r = x - qy

But I am getting stuck in finding y when x is divided by 19. Can someone please help?? I don't have an OA either.

Take LCM of 19 & 11 = 209
Say x = 212

212/11: Quotient = 19 (=y) & remainder = 3

212/19: Quotient = 11 & remainder = 3

19/19: Quotient = 1 & remainder = 0 = Answer = A
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Re: When the positive integer x is divided by 11, the quotient [#permalink]

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10 Mar 2014, 23:31
Can we argue that this is a poorly written question? I was always taught that the quotient was the result of division. At first I paused thinking they must mean just the integer, even though I'd never seen that definition used. Then I decided that must be the "trick" of the question, that y is in fact the answer to x/11.

I got

x = 212

y = 19 + 3/11 = 212/11

y/19 = 212/11 * 1/19 = 212/209 = 1 R 3

The trick would be that people would misinterpret the question and mistake y as 19. But if y is x/11, then we will only have the same remainder when x is a multiple of 19 and 11. (Multiple of 19) + 3 yields a remainder of 3 when 19 is the divisor. Basically, the question would be testing only logic and, in theory, you would never have to figure out any values at all.

So I input the answer as 3, which is not the OA.

Afterward, I looked it up and quotient can also refer to just the integer of the result.

My question: Would this ever be a real GMAT question? We are taught (at least in American schools) that quotient is not just integer, but integer and remainder. Thus, the question appears ambiguous to me and probably a large segment of other GMAT takers. Thoughts?

Edit:

Just consulted OG: "For example, when 28 is divided by 8, the quotient is 3 and the remainder is 4 since 28 = (8)(3) + 4" (p. 108). OG for GRE says something similar. I just find it bizarre that at no time in my formal education did anyone give this definition.

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Re: When the positive integer x is divided by 11, the quotient [#permalink]

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11 Mar 2014, 05:11
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enigma123 wrote:
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

Sol: Given x=11y+3 or y= (x-3)/11
Also x=19a+3 Substitute for x in the above equation we get

y= (19a+3-3)/11 or y=19a/11. Note that "a" is an integer and multiple of 11

y/19 =a/11 where a/11 is an integer and hence remainder is 0

Ans is A
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Re: When the positive integer x is divided by 11, the quotient [#permalink]

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21 Mar 2015, 14:15
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Re: When the positive integer x is divided by 11, the quotient [#permalink]

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26 Jul 2015, 00:20
x=11y+3 and x=19z+3

11y+3=19z+3 => 11y=19z => y/z=19/11

It means that y is a multiple of 19, so when we divide y by 19 we will have r=0

A

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Re: When the positive integer x is divided by 11, the quotient [#permalink]

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15 Aug 2016, 10:26
Hello from the GMAT Club BumpBot!

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Re: When the positive integer x is divided by 11, the quotient [#permalink]

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29 Aug 2016, 10:30
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BOOKMARKED
enigma123 wrote:
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

Guys struggling to solve this. But this is the concept I am trying to apply:

We can extrapolate a general statement from this form. When dividing x by y, the quotient is q and the remainder is r:
x/y = q + r/y

From there, we can solve for x:
x = qy + r (thats the general form of x = 3(integer) + 1)
Or the quotient:
q = x-r/y

Or, even, the remainder itself:
r = x - qy

But I am getting stuck in finding y when x is divided by 19. Can someone please help?? I don't have an OA either.

Please find the solution as attached,
Attachments

File comment: www.GMATinsight.com

Sol5.jpg [ 76.73 KiB | Viewed 5231 times ]

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Re: When the positive integer x is divided by 11, the quotient [#permalink]

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29 Aug 2016, 12:14
x=11*19+3=212
y=quotient of 212/11=19
19/19 gives a remainder of 0

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Re: When the positive integer x is divided by 11, the quotient [#permalink]

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16 Oct 2016, 11:44
A is the correct answer. Here's why:

From the prompt we can derive two equations:

x=11y+3
x=19z+3

From this we can set the two equal to each other, leaving us with...

11y=19z --> manipulate further to give... 11 = (19z)/y --> From this we know that z must equal 11 and y must equal 19 in order for the equation to hold. Therefore, since y = 19, dividing by 19 will give a remainder of 0

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When the positive integer x is divided by 11, the quotient [#permalink]

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06 Nov 2016, 15:34
Hey Everyone.
There are two methods that can be used here as far as i know.

First =>
x=11y+3
and x=19z+3
for some integer z
Hence 11y+3=19z+3
11y=19z
=> z=11y/19
as z needs to be an integer => y must be a multiple of 19.
Hence the remainder that y leaves with 19 must be zero.

Second =>
You see this is a problem solving question.
We will have only one answer.

x=19y+3
x=11z+3
hence combing the equations => x= 3+11*19*p for some integer p.
smallest possible value of x is 3
for the y is zero.
so y=0 is a acceptable value.
what is the remainder when 0 is divided by 19 ?
Its zero.
as every number divides 0.
Hence the answer must be zero.

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Re: When the positive integer x is divided by 11, the quotient [#permalink]

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07 Nov 2016, 13:53
enigma123 wrote:
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

19z + 3 = x = 11y + 3

Or, x = 19*11 + 3 { z = 11 and y = 19 }

or, x = 202

y/19 = Quotient 1 and remainder 0

Hence , answer will be (A) 0

Hope this helps !!

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Re: When the positive integer x is divided by 11, the quotient [#permalink]

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19 Jan 2017, 04:22
Bunuel wrote:
enigma123 wrote:
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

If you decide to go with quotient/remainder formula approach, then I'd suggest to express the info in the stem with it. And then look whether we can somehow manipulate with the expressions at hand to answer the question.

(1) When the positive integer x is divided by 11, the quotient is y and the remainder 3 --> $$x=11y+3$$;
(2) When x is divided by 19, the remainder is also 3 --> $$x=19q+3$$.

Easy to spot that $$19q+3=11y+3$$ --> $$19q=11y$$ --> $$y=\frac{19q}{11}$$. Now as $$y$$ and $$q$$ are integers and 19 is prime then $$\frac{q}{11}$$ must be an integer --> $$y=19*integer$$ --> $$y$$ is a multiple of 19, hence when divide by 19 remainder is 0.

Hope its clear.

Bunuel, what is the relationship between prime and divider in this case?
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Re: When the positive integer x is divided by 11, the quotient [#permalink]

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19 Jan 2017, 05:56
ziyuenlau wrote:
Bunuel wrote:
enigma123 wrote:
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

If you decide to go with quotient/remainder formula approach, then I'd suggest to express the info in the stem with it. And then look whether we can somehow manipulate with the expressions at hand to answer the question.

(1) When the positive integer x is divided by 11, the quotient is y and the remainder 3 --> $$x=11y+3$$;
(2) When x is divided by 19, the remainder is also 3 --> $$x=19q+3$$.

Easy to spot that $$19q+3=11y+3$$ --> $$19q=11y$$ --> $$y=\frac{19q}{11}$$. Now as $$y$$ and $$q$$ are integers and 19 is prime then $$\frac{q}{11}$$ must be an integer --> $$y=19*integer$$ --> $$y$$ is a multiple of 19, hence when divide by 19 remainder is 0.

Hope its clear.

Bunuel, what is the relationship between prime and divider in this case?

We have $$y=\frac{19q}{11}$$. 19 is not a multiple of 11, thus for 19q/11 to be an integer q must be a multiple of 11.
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Re: When the positive integer x is divided by 11, the quotient [#permalink]

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28 Aug 2017, 00:49
Such questions are disaster as they take a lot of time solving. but here is a shorter way i.e through equations.
x/ 11 yields remainder 3 and quotient y. so we can form equation

x= 11y + 3 ........(1)

also we know that when x is divided by 19, remainder is same. let's take "a" as a quotient.

x=19a +3 .........(2)
equating 1 and 2
11y+3= 19a+3
11y=19a

since both 19 and 11 are prime numbers, with no number in common, so the only values a and y can get are
y=19 and a=11

y/19 =19/19 remainder =0

hope it helps.
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Re: When the positive integer x is divided by 11, the quotient   [#permalink] 28 Aug 2017, 00:49
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