The generators of Z/48Z

Find all generators of $\mathbb{Z}/48\mathbb{Z}$ .

Proof. To find the generators, it suffices for us to check whether for an element

a \in \mathbb{Z}/48\mathbb{Z}

that

gcd(a,48) = 1

. In other words, we need to determine the elements of

(\mathbb{Z}/48\mathbb{Z})^{\times}

, where its cardinality is

16

. Obviously,

1

is the generator of

\mathbb{Z}/48\mathbb{Z}

. Notice that

\lvert \mathbb{Z}/48\mathbb{Z} \rvert = 48

, and

48 = 2^4 3

. So we do know that all even elements less or equal to

47

and primes

2

and

3

are not generators. All primes equal to

47

or less are generators of

\mathbb{Z}/48\mathbb{Z}

\begin{align*} 5,7,11,13,17,19,23,29,31,37,41, \text{ and } 47. \end{align*}

So we do have

13

generators, inclusive

1

as the generator so far. What is left are the other

3

elements. We need to check the next odd elements:

\begin{align*} 9,15,21,25,27,33,35,39, \text{ and } 45. \end{align*}

The integers

9,15,21,27,33,39

and

45

can all be divided by

3

. So we have the

3

last generators:

25, 35

and

45

. All together, we get next generators of

\mathbb{Z}/48\mathbb{Z}

\begin{align*} 1,5,7,11,13,17,19,23,25,29,31,35,37,41,45, \text{ and } 47. \end{align*}