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**1. Homework Statement**

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

**2. Homework Equations**

elements 'a' are idempotent if a^2 = a i.e a(a-1)=0

Find the integers 0 < a < 99 (inclusive), such that 100 divides a(a-1)

**3. The Attempt at a Solution**

I know by trial and error that the answers 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 cant do the maths to get a proper reasoning

100 = 2x2x5x5, so does that mean if 100 divides a(a-1) that 2 divides a(a-1) as would 5.

the problem is this wouldnt lead me to any of the known solutions either?

Thanks for any help