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Mod And Near Myriad (Posted on 2011-04-24) Difficulty: 3 of 5
Determine all possible positive integer(s) N < 10,000, such that:

2N = 88 (mod 167), and:

2N = 70 (mod 83)

**** For an extra challenge, solve this problem without using a computer program.

No Solution Yet Submitted by K Sengupta    
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Solution Solution (using excel) | Comment 1 of 4
Well, I wasn't totally up to pencil and paper.

Using Excel, 2^36 = 70 (mod 83) and 2^82 = 1 (mod 83).

So, N = 36 (mod 82)

Also, 2^12 = 88 (mod 167) and 2^83 = 1 (mod 167)

So, N = 12 (mod 83).

The first N which works is 6486.

Since 82 and 83 are relatively prime, and 82*83 = 6806, it follows that
N = 6486 (mod 6806).

The next N that works 6806 + 6486 = 13,292.

Under 10,000, only 6486 works.  

Edited on April 24, 2011, 12:34 pm

Edited on April 24, 2011, 12:37 pm
  Posted by Steve Herman on 2011-04-24 12:25:04

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