Arithmatic, Algebra

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vinayreguri: Junior | Next Rank: 30 Posts; Posts: 29; Joined: Sat May 07, 2011 6:16 am

by vinayreguri » Sun Jun 12, 2011 3:53 am

N is an integer which when divided by 2 leaves a reminder of 1. what will be the reminder when N^2 is divided by 8
A. 1
B. 3
C. 5
D. 7
E. Not a consistent value.

OA: A

Last edited by vinayreguri on Sun Jun 12, 2011 4:13 am, edited 1 time in total.

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Source: Beat The GMAT — Problem Solving | Sun Jun 12, 2011 3:53 am

Frankenstein: Legendary Member; Posts: 1448; Joined: Tue May 17, 2011 9:55 amFollowed by: 53 members; Location: India; Thanked: 375 times

by Frankenstein » Sun Jun 12, 2011 4:06 am

Hi,
N =2k+1.
So, N^2 = (2k+1)^2 = 4k^2+4k+1 = 4k(k+1) + 1
k,(k+1) are 2 consecutive integers. So, one of them is even and other is odd. So, product is even
So, 4k(k+1) = 4*even = multiple of8
So, N^2 when divided by 8 leaves remainder 1

Hence, A

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Geva@EconomistGMAT: GMAT Instructor; Posts: 905; Joined: Sun Sep 12, 2010 1:38 amFollowed by: 123 members; Thanked: 378 times; GMAT Score:760

by Geva@EconomistGMAT » Sun Jun 12, 2011 4:28 am

Frankenstein wrote:Hi,
N =2k+1.
So, N^2 = (2k+1)^2 = 4k^2+4k+1 = 4k(k+1) + 1
k,(k+1) are 2 consecutive integers. So, one of them is even and other is odd. So, product is even
So, 4k(k+1) = 4*even = multiple of8
So, N^2 when divided by 8 leaves remainder 1

Hence, A

Good, although slightly overkill. A few plug ins can show the same pattern. N can only be odd integers:

if N=1, then N^2 = 1: leaves a remainder of 1 from the nearest multiple of 8 (which is zero)
if N=3, then N^2 = 9: leaves a remainder of 1 from the nearest multiple of 8 (which is 9)
if N=5, then N^2 = 25: leaves a remainder of 1 from the nearest multiple of 8 (which is 24)
if N=7, then N^2 = 49: leaves a remainder of 1 from the nearest multiple of 8 (which is 48)

Now it's true that four plug ins do not prove a rule indefinitely - it is possible that somewhere down the line, there is an odd example of N that does not yield a remainder of 1 when divided by 8. But usually we'll see some sort of break from the pattern before reaching 4 consecutive successes. For the purposes of the GMAT, these plug ins are enough to choose A with a high degree of certainty.

Geva
Senior Instructor
Master GMAT
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https://www.mastergmat.com

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cans: Legendary Member; Posts: 1309; Joined: Mon Apr 04, 2011 5:34 amFollowed by: 123 members; Location: India; Thanked: 310 times; GMAT Score:750

by cans » Tue Jun 14, 2011 3:54 am

n=2m+1
n^2 = 4m^2 + 4m + 1
n=1;remainder=1
n=3;remainder=1
n=5;remainder=1
IMO A

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