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When 85 divided by n, the remainder is 7. When 85 divided by

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Senior Manager
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When 85 divided by n, the remainder is 7. When 85 divided by [#permalink] New post 19 Aug 2006, 08:11
When 85 divided by n, the remainder is 7. When 85 divided by 2n, the remainder is n+7. n=?

OA=26
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 [#permalink] New post 19 Aug 2006, 12:47
It would proabably be smarter to use POE here, but here's a quick solution


When 85 is divided by n, the remainder is 7 i.e. 85=nk+7 for some integer k

So nk=78

k could be {1,2,3,6,13,26,39,78}
n could be {78,39,26,13...}

When 85 is divided by 2n, the remainder is n+7

n......2n.......remainder when 85 is divided by 2n

78.....156............85
39...... 78.......... 7 reject
26...... 52.......... 33
13...... 26.......... 7 reject
6........ 12.......... less than 12, reject


I think n could be 78 or 26
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 [#permalink] New post 20 Aug 2006, 01:40
kevincan wrote:
It would proabably be smarter to use POE here, but here's a quick solution


When 85 is divided by n, the remainder is 7 i.e. 85=nk+7 for some integer k

So nk=78

k could be {1,2,3,6,13,26,39,78}
n could be {78,39,26,13...}

When 85 is divided by 2n, the remainder is n+7

n......2n.......remainder when 85 is divided by 2n

78.....156............85
39...... 78.......... 7 reject
26...... 52.......... 33
13...... 26.......... 7 reject
6........ 12.......... less than 12, reject


I think n could be 78 or 26

Exactly same except: I dit not test for n <= 7 because remainder of 85/n is 7 so n can not be less than equal to 7.
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 [#permalink] New post 20 Aug 2006, 09:08
Given
85 = n x m + 7
85 =2nx k + n+ 7

where m and k are two integers

n x m + 7 = 2n x k+ n + 7

or n x m = n x (2k+1)

or n(m-2k+1) = 0

or m = 2k+1

This implies m is odd i.e. multiple of n is odd.

As remainder is 7 in both cases, factoring 85-7 = 78

78 = 2 x 39 = 2x3x13

As we know that m is odd, 78 = 6x13 or 78 = 26x3
where m = 13 or 3 and n = 26 or 6

Eliminate:
For n = 6:
85 = 6x13 +7
85 =12x6 + (6+7) ===> Cannot be as m is odd eliminate n = 6.

For n = 26:
85 = 26x3 + 7
85 = 52x1+ (26+7)

Answer: 26
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 [#permalink] New post 20 Aug 2006, 09:17
As usual thanks Haas
  [#permalink] 20 Aug 2006, 09:17
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When 85 divided by n, the remainder is 7. When 85 divided by

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