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

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

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21 Jan 2012, 00:21
4
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Difficulty:

45% (medium)

Question Stats:

68% (02:01) correct 32% (02:36) wrong based on 658 sessions

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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.

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

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21 Jan 2012, 02:02
27
19
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 is y and th  [#permalink]

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10 Mar 2014, 20:54
10
1
2
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 is y and th  [#permalink]

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21 Jan 2012, 02:13
2
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 is y and th  [#permalink]

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11 Mar 2014, 04:11
2
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 is y and th  [#permalink]

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29 Aug 2016, 09:30
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,
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Sol5.jpg [ 76.73 KiB | Viewed 22847 times ]

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

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07 Nov 2016, 12:53
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 is y and th  [#permalink]

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19 Jan 2017, 03:22
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 is y and th  [#permalink]

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19 Jan 2017, 04:56
1
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 is y and th  [#permalink]

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26 Jan 2018, 13:01
2
Hi All,

We're told that...
X/11 = Y remainder 3
X/19 = something remainder 3

With the first piece of information, we know that X is 3 greater than a multiple of 11; with the second piece of information, we know that X is 3 greater than a multiple of 19. To have a remainder of 3 when you divide X by BOTH 11 and 19, X must be a number that is 3 greater than a MULTIPLE of BOTH 11 and 19.

We're asked what the remainder would be when Y is divided by 19. At this point, you might recognize that you could choose Y=19 and solve from there. If you don't recognize why that that relationship exists, then here's a more step-heavy way to get to the correct answer:

X = (11)(19) + 3 = 209 + 3 = 212

212/11 = Y remainder 3
212 = 11Y + 3
209 = 11Y
209/11 = Y
19 = Y

We're ultimately asked what the remainder would be when 19 is divided by 19. The remainder is 0.

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

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04 Jul 2018, 06:28
1
Top Contributor
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

A fast approach is to find a value of x that meets the given conditions.

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....
Notice that x = 3 meets the above conditions.
3 divided by 11 = 0 with remainder 3. In this case, y = 0
Likewise, 3 divided by 11 also leaves a remainder of 3

What is the remainder when y is divided by 19?
In the above example, y = 0
So, when we divide 0 by 19, the remainder is 0

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

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25 Jul 2018, 16:51
1
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

We can create the following expressions:

x = 11y + 3

and

x = 19Q + 3

Thus:

11y + 3 = 19Q + 3

11y = 19Q

11y/19 = Q

Since Q is an integer and 11 is not divisible by 19, then y must be divisible by 19, and hence the remainder when y is divided by 19 is zero.

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

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24 Apr 2020, 10:04
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.

Hi Bunuel,

I understand the solutions. I just had a doubt regarding y and q. Might be a stupid question. But how do we know that they are integers for sure?
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Re: When the positive integer x is divided by 11, the quotient is y and th  [#permalink]

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24 Apr 2020, 12:07
1
uc26 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.

Hi Bunuel,

I understand the solutions. I just had a doubt regarding y and q. Might be a stupid question. But how do we know that they are integers for sure?

Hi uc26,

The concept of "remainders" ONLY involves integers. For example, if you're dealing with 6/4, then you know that that is 1.5; with remainders though, 6/4 = 1 remainder 2.

When you're dealing with remainders on the GMAT, neither the numerator nor the denominator can involve any decimals (re: you cannot ask for a remainder of 4.7/1.3).

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Re: When the positive integer x is divided by 11, the quotient is y and th   [#permalink] 24 Apr 2020, 12:07