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Idempotents of Z/100Z

  1. Apr 14, 2008 #1
    1. The problem statement, all variables and given/known data
    How do you proove what the idempotents of the commutative ring Z/100Z are?

    2. Relevant 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
  2. jcsd
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