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For what value of 'n' will the remainder of 351^n and 352^n

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CEO
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For what value of 'n' will the remainder of 351^n and 352^n [#permalink] New post 11 Oct 2003, 04:52
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D
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For what value of 'n' will the remainder of 351^n and 352^n be the same when divided by 7?

a) 2
b) 3
c) 6
d) 4
e ) Both 3 and 6


EDIT : 5th option added ..mistake in the problem

Last edited by Praetorian on 13 Oct 2003, 00:39, edited 1 time in total.
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 [#permalink] New post 12 Oct 2003, 23:30
351^n=7p+R1
352^n=7q+R2

351=350+1
352=350+2

n=2) (350+1)*(350+1); R1=1 so, R2=4
n=3) R1=1 R2=1 OK

n=6) R1=1 R2=1 OK
n=4) R1=1 R2=2

two answers
CEO
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 [#permalink] New post 13 Oct 2003, 00:06
The official answer is n=3.

For what value of 'n' will the remainder of 351^n and 352^n be the same when divided by 7?

When 351 is divided by 7, the remainder is 1.
When 352 is divided by 7, the remainder is 2.

Let us look at answer choice (1), n = 2

When 351^2 is divided by 7, the remainder will be 1^2 = 1.
When 352^2 is divided by 7, the remainder will be 2^2 = 4.

So when n = 2, the remainders are different.

When n = 3,

When 351^3 is divided by 7, the remainder will be 1^3 = 1.
When 352^3 is divided by 7, the remainder will be 2^3 = 8.

As 8 is greater than 7, divide 8 again by 7, the new remainder is 1.
So when n = 3, both 351^n and 352^n will have the same remainder when divided by 7.

thanks
praetorian

Last edited by Praetorian on 13 Oct 2003, 00:34, edited 1 time in total.
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 [#permalink] New post 13 Oct 2003, 00:32
My comments

351^N leaves 1 with any positive N, when divided by 7
(350+1)*...*(350+1) all the multiplied members are divisible by 7, save for the last 1.

check out 352=350+2

N=2) (350+2)*(350+2) leaves 4 =7a+4
N=3) (7a+4)*(350+2) leaves 8, or actually 1, =7b+1
N=4) (7b+1)*(350+2) leaves 2, or =7c+2
N=5) (7c+2)*(350+2) leaves 4, or =7d+4
N=6) (7d+4)*(350+2) leaves 8, or actually 1

remainders (from N=2) go in the order of 4, 1, 2, 4, 1, 2, ...

so N=3 and N=6 both match
CEO
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 [#permalink] New post 13 Oct 2003, 00:38
stolyar wrote:
My comments

351^N leaves 1 with any positive N, when divided by 7
(350+1)*...*(350+1) all the multiplied members are divisible by 7, save for the last 1.

check out 352=350+2

N=2) (350+2)*(350+2) leaves 4 =7a+4
N=3) (7a+4)*(350+2) leaves 8, or actually 1, =7b+1
N=4) (7b+1)*(350+2) leaves 2, or =7c+2
N=5) (7c+2)*(350+2) leaves 4, or =7d+4
N=6) (7d+4)*(350+2) leaves 8, or actually 1

remainders (from N=2) go in the order of 4, 1, 2, 4, 1, 2, ...

so N=3 and N=6 both match


i agree

When 351^6 is divided by 7, the remainder will be 1^6 = 1.
When 352^6 is divided by 7, the remainder will be 2^6 = 64.

But 64 > 7 ,so divide 64/ 7 , remainder is 1.

Both 3 and 6 work...

let me make necessary changes to the problem

thanks
praetorian
  [#permalink] 13 Oct 2003, 00:38
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For what value of 'n' will the remainder of 351^n and 352^n

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