Find the Remainder

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melguy: Master | Next Rank: 500 Posts; Posts: 335; Joined: Mon Mar 21, 2011 11:31 pm; Location: Australia / India; Thanked: 37 times; Followed by:2 members

Find the Remainder

by melguy » Fri Jun 21, 2013 4:48 am

Please help me with the problem. I understand that we need to apply the formula DIVIDEND = (DIVISOR X QUOTIENT) + REMAINDER but I am unable to solve. Thanks

A number X when divided by 289 leaves 18 as the remainder. The same number when divided by 17 leaves Y as the remainder. Find Y.

a 1
b 2
c 11
d 20
e 23

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Source: Beat The GMAT — Problem Solving | Fri Jun 21, 2013 4:48 am

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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 » Fri Jun 21, 2013 5:14 am

melguy wrote:
A number X when divided by 289 leaves 18 as the remainder. The same number when divided by 17 leaves Y as the remainder. Find Y.

a 1
b 2
c 11
d 20
e 23

When it comes to remainders, we have a nice rule that says:

If N divided by D, leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.

For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

So, . . . A number X when divided by 289 leaves 18 as the remainder
Possible values of X: 18, 18+(289), 18+(2)(289), 18+(3)(289). . .

Of course, we don't need to find lots of possible values of X. We need only one possible value. So, let's take the smallest, which is 18.

If X = 18, then when we divide 18 by 17, we get remainder 1.

Answer: A

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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GMAT/MBA Expert

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 » Fri Jun 21, 2013 5:23 am

melguy wrote:Please help me with the problem. I understand that we need to apply the formula DIVIDEND = (DIVISOR X QUOTIENT) + REMAINDER but I am unable to solve. Thanks

Another approach is to use the rule you're referring to.
The rule says, "If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2

So, . . . A number X when divided by 289 leaves 18 as the remainder.
Let's add one extra piece of information to this: When X is divided by 289, we get Q with remainder 18
So, using our rule above, we can write: X = 289Q + 18 (for some integer Q)

The same number when divided by 17 leaves Y as the remainder.
In other words, what is the remainder when X is divided by 17?
Well, X = 289Q + 18
Rewrite: X = 289Q + 17 + 1
Factor out 17 from red part: X = 17(17Q + 1) + 1
In other words, X = 17(something) + 1
Here, we can see that if we divide X by 17, the remainder will be 1

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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topspin20: Junior | Next Rank: 30 Posts; Posts: 25; Joined: Mon Jul 29, 2013 10:17 am

by topspin20 » Tue Jul 30, 2013 3:28 am

Hi Brent,

This is very helpful, but I have one question about the following:

'If N divided by D, leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.'

I'm not familiar with calculating remainders when the divisor is larger than the dividend, i.e. 18 divided by 289.

Thank you!

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pm; Location: New York, NY; Thanked: 13060 times; Followed by:1906 members; GMAT Score:790

by GMATGuruNY » Tue Jul 30, 2013 5:34 am

melguy wrote:Please help me with the problem. I understand that we need to apply the formula DIVIDEND = (DIVISOR X QUOTIENT) + REMAINDER but I am unable to solve. Thanks

A number X when divided by 289 leaves 18 as the remainder. The same number when divided by 17 leaves Y as the remainder. Find Y.

a 1
b 2
c 11
d 20
e 23

A number X when divided by 289 leaves 18 as the remainder.
In other words, X is 18 more than a multiple of 289.
Translated into math:
X = 289a + 18, where a is an integer greater than or equal to 0.
If a = 0, then X = 289*0 + 18 = 18.
If a = 1, then X = 289*1 + 18 = 307.
If a = 2, then X = 289*2 + 18 = 596.

Note the following:
The LEAST POSSIBLE VALUE Is the REMAINDER ITSELF (18).
To generate other possible values for X, just add the divisor (289) to each resulting value:
X = 18, 18+289 = 307, 307+289 = 596...

The same number when divided by 17 leaves Y as the remainder.
The least possible value for X is 18.
If X = 18, then X/17 = 18/17 = 1 R 1.

Since R=1, the correct answer is A.

Other remainder problems:
https://www.beatthegmat.com/positive-int ... 96845.html
https://www.beatthegmat.com/remainder-of ... 54799.html
https://www.beatthegmat.com/remainder-t115616.html
https://www.beatthegmat.com/remainder-t187461.html

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GMAT/MBA Expert

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 » Tue Jul 30, 2013 6:07 am

topspin20 wrote:Hi Brent,

This is very helpful, but I have one question about the following:

'If N divided by D, leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.'

I'm not familiar with calculating remainders when the divisor is larger than the dividend, i.e. 18 divided by 289.

Thank you!

18 divided by 289 equals 0 with remainder 18
In other words, 289 divides into 18 zero times with remainder 18.

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pm; Location: East Bay all the way; Thanked: 625 times; Followed by:119 members; GMAT Score:780

by Matt@VeritasPrep » Tue Jul 30, 2013 8:28 am

topspin20 wrote:Hi Brent,

This is very helpful, but I have one question about the following:

'If N divided by D, leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.'

I'm not familiar with calculating remainders when the divisor is larger than the dividend, i.e. 18 divided by 289.

Thank you!

Assuming we're working with positive integers, if the divisor is larger than the dividend, the dividend = the remainder.

For instance, 18 / 289 = 0 with a remainder of 18. This is a funny property of remainders that the GMAT has used in the past, so it's a good thing to ask about!

To see this is the formula given:

Dividend = Divisor*Quotient + Remainder

18 = 289*0 + 18

So the dividend is 18, the divisor is 289, the quotient is 0, and the remainder is 18.

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