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When the positive integer x is divided by 11

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When the positive integer x is divided by 11

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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

The OA is the option A.

I am struggling to solve this PS question. Could someone give me a clarification here? Please.

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Hello Vjesus12.

Let's take a look at your question.

When the positive integer x is divided by 11, the quotient is y and the remainder 3, this implies that $$x=11\cdot y+3.$$ When x is divided by 19, the remainder is also 3 $$x=19\cdot k+3.$$

Using these two equations we get $$11\cdot y+3=19\cdot k+3\ \ \Rightarrow\ \ 11y=19k$$ $$\Rightarrow\ \ y=\frac{19k}{11}$$ Now, since y is an integer and 19 is not divisible by 11, then we have that k must be divisible by 11 (in order to get an integer value in the right side). So, $$\Rightarrow\ \ y=19\cdot\frac{k}{11}=19\left(integer\right)$$ This implies that y is multiple of 19.

In conclusion, y is divisible by 19 and therefore the remainder when y is divided by 19 is 0.

This is why the correct answer is the option A.

I hope it helps.

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Since the number x gives remainder 3 for both 11 and 19, we can say that x-3 is multiple of 11 and 19 both.
So, y which we get when we divide x by 11 is same when we divide x-3 by 11

=> y is a multiple of 19, hence remainder will be 0

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GMAT/MBA Expert

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VJesus12 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

Answer: A

Cheers,
Brent

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VJesus12 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.

Answer: A

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Jeffrey Miller Head of GMAT Instruction

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Hi VJesus12,

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.

Final Answer: A

GMAT assassins aren't born, they're made,
Rich

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Contact Rich at Rich.C@empowergmat.com

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