# Thread: Is this the correct...?

1. ## Is this the correct...?

Is this the correct cayley table for the units of integers mod 21?

how would you find the order of each element?

2. Originally Posted by universalsandbox
how would you find the order of each element?
What ye did looks correct.

To find the order of, say $\displaystyle [5]_{21}$ you need to take exponents until you reach $\displaystyle [1]_{21}$.

$\displaystyle [5]_{21}^2 = [4]_{21}$, $\displaystyle [5]_{21}^3 = [20]_{21}$, $\displaystyle [5]_{21}^4 = [16]_{21}$, $\displaystyle [5]_{21}^5 = [17]_{21}$, $\displaystyle [5]_{21}^6 = [1]_{21}$.

Therefore the order of $\displaystyle [5]_{21}$ is $\displaystyle 6$.

3. I think I've got it now.

$\displaystyle [16]_{21} : [16]_{21}^2 = [4]_{21}, [16]_{21}^3 = [1]_{21}$

Order 3

This seems very tedious. Is there a faster way (by hand)?

4. Originally Posted by universalsandbox
This seems very tedious. Is there a faster way (by hand)?
Yes, we can show the order of an element must divide $\displaystyle \phi(n)$. In this case $\displaystyle \phi(20) = 8$.

Therefore, the only exponent you need to check are $\displaystyle 2,4,8$. Instead of all of them.

5. I believe in this case

phi(21) = 12

Note: There are 12 units.

So the orders must be 1, 2, 3, 4, 6 or 12.

Ponder this statement: To find the order of the element it is suffice to check only it's 2nd and 3rd power.

Why?