The quadratic equation x^2-3x+2=0 has the "correct" number of solutions modulo 5 and 7. However, modulo 6 the equation has four solutions; namely, 1, 2, 4, and 5. For what positive integers n does the equation x^2-3x+2=0 have exactly two incongruent solutions modulo n?

(In reply to re: Proof by McWorter)

Supposing that n=ab, and a and b are coprime, then ac-bd=+-1 for positive integers c,d. By the theorem that Ax-By=1 is solvable for the integers x,y if and only if A and B are coprime, if x=X, y=Y is a solution, then X and B are coprime (as are X and Y, and A and Y). Hence c cannot be a multiple of b nor can d be a multiple of a. Hence ac and bd are consecutive numbers, and neither is a multiple of n=ab. Hence ac and bd are an x-1 and an x-2 in some order for an x that is neither 1 nor 2 mod n=ab.

Fun problem, McW!

Edited on September 3, 2005, 11:37 pm

Posted by Richard on 2005-09-03 23:09:09