Let $\varphi(n)$ denote Euler’s totient function, which counts the number of $d$ less than $n$ such that $d \not\mid n$.

  1. Compute $\varphi(8400)$.
  2. Prove that $\varphi(n)$ must be even for all $n > 2$.
  3. Show that there is no positive integer $n$ such that $\varphi(n)=14$.
  4. For which positive integers $n$ does $\varphi(n) | n$?
  5. Recall that a number $n$ is perfect if the sum of its proper divisors is $n$. Since $n$ is the only non-proper divisor of $n$, this is equivalent to saying a number $n$ is perfect if the sum of all its divisors is $2n$.

  6. Can a power of a prime number be perfect?
  7. Show that if $n$ is a perfect number, the sum of the reciprocals of all the divisors of $n$ is $2$.