State the number of integers which have an order modulo $n$ for
(a) $n = 20$
(b) $n = 200$
(c) $n = 2000$
(d) $n = 20 000$
The number of integers that have an order modulo $n$ is those from 1 to $n$ which are co-prime to $n$, that is $\phi(n)$.
(a) $\phi(20) = 20 \times (1-\frac{1}{2}) \times (1-\frac{1}{5}) = 8$
(b) $\phi(200) = 200 \times (1-\frac{1}{2}) \times (1-\frac{1}{5}) = 80$
(c) $\phi(2000) = 2000 \times (1-\frac{1}{2}) \times (1-\frac{1}{5}) = 800$
(d) $\phi(20000) = 20000 \times (1-\frac{1}{2}) \times (1-\frac{1}{5}) = 8000$