Exponent Issues

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AIM TO CRACK GMAT: Senior | Next Rank: 100 Posts; Posts: 80; Joined: Sat Sep 15, 2012 1:07 amFollowed by: 1 member; Thanked: 1 times

by AIM TO CRACK GMAT » Sun Oct 20, 2013 6:31 am

What is the remainder when 1027199 is divided by 5?What is the remainder when 1027^199 is divided by 5?

a) 1
b) 2
c) 3
d) 4

[spoiler]OA = C[/spoiler]

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theCodeToGMAT: Legendary Member; Posts: 1556; Joined: Tue Aug 14, 2012 11:18 pmFollowed by: 34 members; Thanked: 448 times; GMAT Score:650

by theCodeToGMAT » Sun Oct 20, 2013 6:38 am

(1027)^199/5

(2)^199/5 [as 1027/5 leaves remainder "2"]

199/4 = Remainder = 3

So, rewrite (2)^3/5

8/5 = 3

Answer [spoiler]{C}[/spoiler]

R A H U L

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theCodeToGMAT: Legendary Member; Posts: 1556; Joined: Tue Aug 14, 2012 11:18 pmFollowed by: 34 members; Thanked: 448 times; GMAT Score:650

by theCodeToGMAT » Sun Oct 20, 2013 6:42 am

Another Approach

(2)^199/5 [as 1027/5 leaves remainder "2"]

{[(2)^4]49 * 2^3}/5 [break 2 in such a power which yields remainder "1" when divided by 5]

As, (2^4)/5 leaves remainder 1

So,

(1)^49 * 2^3/5
= 8/5 = 3

R A H U L

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mevicks: Master | Next Rank: 500 Posts; Posts: 269; Joined: Thu Sep 19, 2013 12:46 amFollowed by: 7 members; Thanked: 94 times

by mevicks » Sun Oct 20, 2013 6:43 am

AIM TO CRACK GMAT wrote:What is the remainder when 1027^199 is divided by 5?

a) 1
b) 2
c) 3
d) 4

[spoiler]OA = C[/spoiler]

Cyclicity of 7 is 4 (7,9,3,1). Thus considering only the units digit of 1027 we can find the remainder.
units digit of 7^200 is 1
thus units digit of 7^199 would be 3
Any number having 3 as the units digit when divided by 5 would yield a remainder of 3.
eg: 13/5 , 113/5 ..

[spoiler]Answer C : 3[/spoiler]

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Uva@90: Master | Next Rank: 500 Posts; Posts: 490; Joined: Thu Jul 04, 2013 7:30 amFollowed by: 5 members; Location: Chennai, India; Thanked: 83 times

by Uva@90 » Sun Oct 20, 2013 6:46 am

AIM TO CRACK GMAT wrote:What is the remainder when 1027^199 is divided by 5?

a) 1
b) 2
c) 3
d) 4

[spoiler]OA = C[/spoiler]

Hi,
Last Digit of any number is sufficient to find whether the number is divisible by 5 or not(or to find the reminder)

So 1027^199 can be rephrased as 7^199

Now note the patter of positive powers of 7 below,

7^1 ends in 7
7^2 ends in 9(7*7=49)
7^3 ends in 3(9*7)
7^4 ends in 1(3*7)

So series goes like this 7,9,3,1,7,9,3,1...........

We are asked to find 7^199.
You can note that series repeats after every 4 number.

Divide 199/4 gives the remainder 3 so 7^199 = 7^3
7^3 End with 3

Hence Answer is C

Regards,
Uva.

Known is a drop Unknown is an Ocean

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ganeshrkamath: Master | Next Rank: 500 Posts; Posts: 283; Joined: Sun Jun 23, 2013 11:56 pmFollowed by: 26 members; Location: Bangalore, India; Thanked: 97 times; GMAT Score:750

by ganeshrkamath » Sun Oct 20, 2013 6:48 am

AIM TO CRACK GMAT wrote:What is the remainder when 1027199 is divided by 5?What is the remainder when 1027^199 is divided by 5?

a) 1
b) 2
c) 3
d) 4

[spoiler]OA = C[/spoiler]

1027 mod 5 = 2
2^199 mod 5 = ((2^2)^99 * 2) mod 5
= (4^99 * 2) mod 5
= ((-1)^99 * 2) mod 5
= (-1 * 2) mod 5
= (-2) mod 5
= 3

Choose c

Cheers

Every job is a self-portrait of the person who did it. Autograph your work with excellence.

Kelley School of Business (Class of 2016)
GMAT Score: 750 V40 Q51 AWA 5 IR 8
https://www.beatthegmat.com/first-attemp ... tml#688494

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

by Brent@GMATPrepNow » Sun Oct 20, 2013 6:55 am

Just want to mention two things.

1. ganeshrkamath and theCodeToGMAT are using something called "modular arithmetic" to solve this question. While this works, the GMAT does not require you to know this for the test.

2. Uva@90 and mevicks are using a different approach that does not require modular arithmetic. This strategy is covered in this free video: https://www.gmatprepnow.com/module/gmat- ... ts?id=1031

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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theCodeToGMAT: Legendary Member; Posts: 1556; Joined: Tue Aug 14, 2012 11:18 pmFollowed by: 34 members; Thanked: 448 times; GMAT Score:650

by theCodeToGMAT » Sun Oct 20, 2013 6:56 am

AIM TO CRACK GMAT wrote:What is the remainder when 1027199 is divided by 5?What is the remainder when 1027^199 is divided by 5?

a) 1
b) 2
c) 3
d) 4

[spoiler]OA = C[/spoiler]

AIM TO CRACK GMAT, although you have received solutions for the problem, I would advise you to correct the question; this will help to maintain the quality of the post.

Thanks!

R A H U L

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