# Idempotents of the ring Z/100Z

## Main Question or Discussion Point

How do you proove what the idempotents of the ring Z/100Z are?

I know by trial and error that they are the elements 0,1,25,76 but have no proof as to why this is.
i know i have to find the integers 'a' such that 100 divides a(a-1)=a^2 -a but can do the maths to get a proper reasoning

any help?
thanks

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quasar987
Chose an element [a] of Z/100Z whose representant a is such that $0\leq a \leq 99$ (for simplicity). Then by definition, [a] will be idempotent if [a]²=[a]. That is to say, if a²+100Z = a+100Z. But this will happen if and only if there exists a k such that a² = a + k. Can you go from there?