Find the smallest positive integer x for which (7x²⁵ - 10)/83 is an integer.

Can you do this in a short time using pen and paper, and eventually a hand calculator, but no computer programs?

(In reply to thoughts by Charlie)

I did much the same (except your determination of 37 is more elegant than what I did)

I then constructed a table of x^5 mod 83  (I have to admit to cheating and using google calculator to compute the values for the table.  This yielded 16^5 mod 83 = 37 and 69^5 mod 83 = 16 so the answer is 69. 

The second half of the table (42-82) can be easily computed as x^5 mod 83 = 83 - ((83-x)^5 mod 83)
because x^5 mod 83 = (x-83)^5 mod 83 = -(83-x)^5 mod 83 = 83-((83-x)^5 mod 83)

Anyway, the table only requires 41 x^5 mod 83 calcs and if you build from the ends in you caould stop at 16 such calcs

Posted by Joel on 2006-12-13 09:36:04