Exercises - The Order of an Integer and Primitive Roots

  1. Find $\text{ord}_{257} 5$.

  2. Show that $2$ is a primitive root of $11$.

  3. How many incongruent primitive roots does 14 have?

  4. Suppose $n$ is a positive integer, and $a^{-1}$ is a multiplicative inverse of $a\pmod{n}$.

    1. Show $\textrm{ord}_n a = \textrm{ord}_n (a^{-1})$
    2. If $a$ is a primitive root modulo $n$, must $a^{-1}$ also be a primitive root?
  5. Show if $n$ is a positive integer relatively prime to integer $a$ and $\textrm{ord}_{n} a = st$, then $\textrm{ord}_{n} a^t = s$.

  6. Show if $\gcd (a,n) = \gcd (b,n) = \gcd (\textrm{ord}_n a, \textrm{ord}_n b) = 1$, then $\textrm{ord}_n (ab) = (\textrm{ord}_n a)(\textrm{ord}_n b)$.  

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