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

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BTGmoderatorDC: Moderator; Posts: 7187; Joined: Thu Sep 07, 2017 4:43 pm; Followed by:23 members

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

by BTGmoderatorDC » Tue Jul 02, 2019 8:36 pm

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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. 4
B. 3
C. 2
D. 1
E. 0

OA E

Source: Manhattan Prep

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Source: Beat The GMAT — Problem Solving | Tue Jul 02, 2019 8:36 pm

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by Ian Stewart » Wed Jul 03, 2019 4:13 am

There's an old GMATPrep question that is almost identical to this one. Since the value of x can be 3, the quotient can be 0 (when we divide 3 by 11, the quotient is zero and the remainder is 3), so if one of the five answers is right, it has to be E.

Or, if one wants a proper 'proof', then the information in the question tells us x is equal to 11y + 3, and also to 19q + 3 (where q is the quotient when we divide x by 19). So these must equal each other:

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

But if 11y and 19q are the same number, they must have the same divisors, and since 19 is clearly a divisor of 19q, it must also be a divisor of 11y. It's not a divisor of 11, so it must be a divisor of y, and the remainder is thus zero when we divide y by 19.

For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pm; Location: Vancouver, BC; Thanked: 5254 times; Followed by:1268 members; GMAT Score:770

by Brent@GMATPrepNow » Wed Jul 03, 2019 6:02 am

BTGmoderatorDC 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. 4
B. 3
C. 2
D. 1
E. 0

OA E

Source: Manhattan Prep

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: E

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8086; Joined: Sat Apr 25, 2015 10:56 am; Location: Los Angeles, CA; Thanked: 43 times; Followed by:29 members

by Scott@TargetTestPrep » Sat Jul 06, 2019 5:08 pm

BTGmoderatorDC 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. 4
B. 3
C. 2
D. 1
E. 0

OA E

Source: Manhattan Prep

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: E

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