Question

Find all the zero divisors in

(a)${\mathrm{\mathbb{Z}}}_7$

(b)${\mathrm{\mathbb{Z}}}_8$

(c)role="math" localid="1646372935683" ${\mathrm{\mathbb{Z}}}_9$

(d)role="math" localid="1646372944841" ${\mathrm{\mathbb{Z}}}_{10}$

Short answer

(a) No zero divisors

(b)$\left[2\right],\left[4\right],\left[6\right]$

(c)$\left[3\right],\left[6\right]$

(d)$\left[2\right],\left[4\right],\left[5\right],\left[6\right],\left[8\right]$

Step-by-step solution

Find the units 

Make the Multiplication table as:

$\odot$	[0]	[1]	[2]	[3]	4]	[5]	[6]
[0]	[0]	[0]	[0]	[0]	[0]	[0]	[0]
[1]	[0]	[1]	[2]	[3]	[4]	[5]	[6]
[2]	[0]	[2]	[4]	[6]	[1]	[3]	[5]
[3]	[0]	[3]	[6]	[2]	[5]	[1]	[4]
[4]	[0]	[4]	[1]	[5]	[2]	[6]	[3]
[5]	[0]	[5]	[3]	[1]	[6]	[4]	[2]
[6]	[0]	[6]	[5]	[4]	[3]	[2]	[1]

Therefore, there are no zero divisors as 7 is a prime number.

Find the units

Make the Multiplication table as:

$\odot$	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]
[0]	[0]	[0]	[0]	[0]	[0]	[0]	[0]	[0]
[1]	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]
[2]	[0]	[2]	[4]	[6]	[0]	[2]	[4]	[6]
[3]	[0]	[3]	[6]	[1]	[4]	[7]	[2]	[5]
[4]	[0]	[4]	[0]	[4]	[0]	[4]	[0]	[4]
[5]	[0]	[5]	[2]	[7]	[4]	[1]	[6]	[3]
[6]	[0]	[6]	[4]	[2]	[0]	[6]	[4]	[2]
[7]	[0]	[7]	[6]	[5]	[4]	[3]	[2]	[1]

From the table, it is observed that role="math" localid="1646377486571" $\left[2\right]\odot \left[4\right]=\left[0\right]$ and $\left[3\right]\odot \left[6\right]=\left[0\right]$.

Therefore, the zero divisors are$\left[3\right],\left[6\right]$ .

Find the units

Make the Multiplication table as:

$\odot$	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]
[0]	[0]	[0]	[0]	[0]	[0]	[0]	[0]	[0]	[0]
[1]	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]
[2]	[0]	[2]	[4]	[6]	[8]	[1]	[3]	[5]	[7]
[3]	[0]	[3]	[6]	[0]	[3]	[6]	[0]	[3]	[6]
[4]	[0]	[4]	[8]	[3]	[7]	[2]	[6]	[1]	[5]
[5]	[0]	[5]	[1]	[6]	[2]	[7]	[3]	[8]	[4]
[6]	[0]	[6]	[3]	[0]	[6]	[3]	[0]	[6]	[3]
[7]	[0]	[7]	[5]	[3]	[1]	[8]	[6]	[4]	[2]
[8]	[0]	[8]	[7]	[6]	[5]	[4]	[3]	[2]	[1]

From the table, it is observed that $\left[3\right]\odot \left[3\right]=\left[0\right]$ and $\left[3\right]\odot \left[6\right]=\left[0\right]$.

Therefore, the zero divisors are $\left[3\right],\left[6\right]$.

Find the units

Make the Multiplication table as:

$\odot$	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]	[9]
[0]	[0]	[0]	[0]	[0]	[0]	[0]	[0]	[0]	[0]	[0]
[1]	[0]	[1]	[2]	[3]	[4]	[5]	[6]	[7]	[8]	[9]
[2]	[0]	[2]	[4]	[6]	[8]	[0]	[2]	[4]	[6]	[8]
[3]	[0]	[3]	[6]	[9]	[2]	[5]	[8]	[1]	[4]	[7]
[4]	[0]	[4]	[8]	[2]	[6]	[0]	[4]	[8]	[2]	[6]
[5]	[0]	[5]	[0]	[5]	[0]	[5]	[0]	[5]	[0]	[5]
[6]	[0]	[6]	[2]	[8]	[4]	[0]	[6]	[2]	[8]	[4]
[7]	[0]	[7]	[4]	[1]	[8]	[5]	[2]	[9]	[6]	[3]
[8]	[0]	[8]	[6]	[4]	[2]	[0]	[8]	[6]	[4]	[2]
[9]	[0]	[9]	[8]	[7]	[6]	[5]	[4]	[3]	[2]	[1]

From the table, it is observed that $\left[2\right]\odot \left[5\right]=\left[0\right]$, $\left[4\right]\odot \left[5\right]=\left[0\right],\left[6\right]\odot \left[5\right]=\left[0\right]$, and $\left[8\right]\odot \left[5\right]=\left[0\right]$.

Therefore, the zero divisors are$\left[2\right],\left[4\right],\left[5\right],\left[6\right],\left[8\right]$.