Number system

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goyalsau: Legendary Member; Posts: 866; Joined: Mon Aug 02, 2010 6:46 pm; Location: Gwalior, India; Thanked: 31 times

by goyalsau » Mon Nov 15, 2010 2:42 am

HI! Friends,

Yesterday one of my friend asked one question. It is from Number Properties. Please share your views

One number is {159 * 66 * 99 * 135 * 29 * 34 } What we should subtract from the number to make it divisible by 31 ?

I don't remember the options exactly,

they were like

11
9
3
4

Or may be i am not posting the right options. but i just want to know how to approach these kind of questions. I am completely confused.............

Saurabh Goyal
[email protected]
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EveryBody Wants to Win But Nobody wants to prepare for Win.

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Source: Beat The GMAT — Problem Solving | Mon Nov 15, 2010 2:42 am

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Rahul@gurome: GMAT Instructor; Posts: 1179; Joined: Sun Apr 11, 2010 9:07 pmFollowed by: 88 members; Location: Milpitas, CA; Thanked: 447 times

by Rahul@gurome » Mon Nov 15, 2010 3:40 am

goyalsau wrote:HI! Friends,

Yesterday one of my friend asked one question. It is from Number Properties. Please share your views

One number is {159 * 66 * 99 * 135 * 29 * 34 } What we should subtract from the number to make it divisible by 31 ?

I don't remember the options exactly,

they were like

11
9
3
4

Or may be i am not posting the right options. but i just want to know how to approach these kind of questions. I am completely confused.............

We have to use modular arithmetic here. According to modular arithmetic, (a mod b) = c means when a is divided by b, the remainder is c. A fundamental formula in modular arithmetic is

(ab) mod c = [a mod c * b mod c] mod c

Say, N = 159*66*99*135*29*34
Therefore, (N mod 31) = [{(159 mod 31)*(66 mod 31)*(99 mod 31)*(135 mod 31)*(29 mod 31)*(34 mod 31)} mod 31]

Now,

159 mod 31 = 4
66 mod 31 = 4
99 mod 31 = 6
135 mod 31 = 11
29 mod 31 = 29
34 mod 31 = 3

Thus, (N mod 31)
= [{4*4*6*11*29*3} mod 31]
= [{32*99*29} mod 31]
= [{(32 mod 31)*(99 mod 31)*(29 mod 31)} mod 31]
= [{1*6*29} mod 31]
= 174 mod 31
= 19

Therefore we must subtract 19 from N to make it divisible by 31.

Rahul Lakhani
Quant Expert
Gurome, Inc.
https://www.GuroMe.com
On MBA sabbatical (at ISB) for 2011-12 - will stay active as time permits
1-800-566-4043 (USA)
+91-99201 32411 (India)

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beat_gmat_09: Master | Next Rank: 500 Posts; Posts: 437; Joined: Sat Nov 22, 2008 5:06 amFollowed by: 1 member; Location: India; Thanked: 50 times; GMAT Score:580

by beat_gmat_09 » Mon Nov 15, 2010 4:19 am

Is this a GMAT question ? Source ?

Hope is the dream of a man awake

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goyalsau: Legendary Member; Posts: 866; Joined: Mon Aug 02, 2010 6:46 pm; Location: Gwalior, India; Thanked: 31 times

by goyalsau » Mon Nov 15, 2010 4:44 am

beat_gmat_09 wrote:Is this a GMAT question ? Source ?

I don't know buddy one of my friend asked me this question...

Rahul has provided a splendid solution, But the thing is { its very hard to remember formula } approach is good but i don't know will i able to remember it when i will be asked the same question again..........

Saurabh Goyal
[email protected]
-------------------------

EveryBody Wants to Win But Nobody wants to prepare for Win.

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goyalsau: Legendary Member; Posts: 866; Joined: Mon Aug 02, 2010 6:46 pm; Location: Gwalior, India; Thanked: 31 times

by goyalsau » Mon Nov 15, 2010 5:16 am

Rahul thanks for the Wonderfull Solution.

can we use this property to solve one more question.

76 ^ 47 is the number and we have to make it divisible by 47

29 ^ 47 mode 47

it can we written as 841 ^ 23 * 29 { 47 * 17 = 799 }

42 ^ 23 * 29 mode 47

1764 ^ 11 * 29 * 42 mode 47

25 ^ 11 * 29 * 42 mode 47

525 ^5 * 29 * 25 * 42 mode 47

8 ^ 5 * 29 * 25 * 42 mode 47

64 ^ 2 * 8 * 29 * 25 * 42 mode 47

27 ^ 2 * 8 * 29 * 25 * 42 mode 47

729 * 8 * 29 * 25 * 42 mode 47

24 * 8 * 29 * 25 * 42 mode 47

192 * 725 * 42 mode 47

4 * 20 * 42 mode 47

80 * 42 mode 47

33 * 42 mode 47

462 * 3 mode 47

39 * 3 mode 47

117 mode 47

23 mode 47

we need to subtract 23 from 76 ^ 47

Saurabh Goyal
[email protected]
-------------------------

EveryBody Wants to Win But Nobody wants to prepare for Win.

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