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

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

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

Please find the solution as attached,

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 !!

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.

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.

Final Answer:

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

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

RELATED VIDEO FROM OUR COURSE

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

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).

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

As per the given question stem,

y should be a multiple of 19

And, x will be equal to 19*11*a + 3 (a being a random integer)

Hence, answer is 0.

OA: A

Given: When the positive integer x is divided by 11, the quotient is y and the remainder 3
therefore: x=11y + 3
Keep in note, both x and y here are integers

Given: When x is divided by 19, the remainder is also 3 (lets call the quotient q)
x = 19q+3

Manipulating this expression
(x-3)/19 = q
We know x = 11y+3. Substituting for x in the above equation
11y+3-3 /19 = q
11y/19 = q
We know that y is an integer and 11 is not divisible by 19
So since it can be any integer, it could be 19
Hence 11x19/19 = 11 with remainder 0
Hence A