Find Idempotent Element in Z/mnZ (m,n Relatively Prime)

[mansi]

here's a real tough one ( at least for me) ...show that the ring Z/mnZ where m ,n are relatively prime has an idempotent element other than 0 and 1.
i looked at examples and it works...
do we look for solutions of the equation a^2 -a = kmn , for some k in Z( that is, other than 0 and 1)?
help!

 

[matt grime]

m and n are coprime. The only thing you know about coprime integers is that there are numbers a and b such that am+bn=1. What can you conclude now?

 

[mansi]

ok...so am + bn= 1 implies...1-bn = am
that implies... bn(1-bn) = abmn = 0...
so bn is an idempotent element ...
i looked at " am +bn =1" a hundred times before posting this question...but it flashed just now! thanks a ton!