# Which elements of z42 are invertibles?

• duki

## Homework Statement

which elements of z42 are invertibles?

## The Attempt at a Solution

In my notes I have that invertibles are relatively prime to the order of the group. So I have

1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41.

Is that the case for all groups?

## Homework Statement

which elements of z42 are invertibles?

## The Attempt at a Solution

In my notes I have that invertibles are relatively prime to the order of the group. So I have

1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41.

Is that the case for all groups?

Is what the case for all groups?

Interesting fact: Note that 1 is its own inverse. That leaves 11 invertibles, so if you repeatedly remove pairs of invertibles, you have at least one left without a partner, meaning it has to be its own inverse. Which one is it? Is there an easy way to spot it? Is there more than one?

Is what the case for all groups?
Can you always take the relatively primes and get the invertibles?

Can you always take the relatively primes and get the invertibles?

Yes, but do you know why? Hint: can you apply Fermat's little theorem?