Getting phi of a large number

  1. phi(20^100)
    = phi(4^100 * 5^100)
    = phi(2^200 * 5^100)
    = (2^200 - 2^199)(5^100 - 5^99)
    = 2^199(2-1) * 5^99(5-1)
    = 2^199 * 5^99 * 4
    = 2^201 * 5^99.

    I don't understand line 4-7. Can anyone explain?
     
  2. jcsd
  3. CRGreathouse

    CRGreathouse 3,682
    Science Advisor
    Homework Helper

    phi(p^a) = (p-1)(p^(a-1)) for p prime
     
  4. also since phi is multiplicative phi(a*b) = phi(a)*phi(b) when gcd(a,b)=1
     
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