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**