Find all zero divisors of z_81

To find all zero divisors of $z_{81}$, we first need to understand what $z_{81}$ is. $z_{81}$ is the ring of integers modulo 81. This means that we are working with integers, but with the additional rule that any multiple of 81 is equivalent to 0.

Now, a zero divisor is an element of the ring that is not equal to 0, but whose product with another element is equal to 0. In other words, if a and b are elements of the ring, and $a\cdot b=0$, then either a or b (or both) is a zero divisor.

To find all zero divisors of $z_{81}$, we can simply check each element of the ring, and see if it has any factors of 3 or 9 in common with 81. If it does, then it is a zero divisor.

Let's do this systematically. The elements of $z_{81}$ are:
0, 1, 2, 3, ..., 77, 78, 79, 80

We can see that 81 is a multiple of $3^4$, so any element with a factor of $3^2$ (i.e., any element that is divisible by 9) will be a zero divisor.

Therefore, the zero divisors of $z_{81}$ are:
0, 9, 18, 27, 36, 45, 54, 63, 72, 81 (which is equivalent to 0)

And that's it! We have found all the zero divisors of $z_{81}$.