Question

Prove or disprove: If$\left[a\right]⊝\left[b\right]=\left[0\right]$in${\mathrm{\mathbb{Z}}}_n$, then$\left[a\right]=\left[0\right]$ or $\left[b\right]=\left[0\right]$.

Short answer

The given result is false. It is proved that when$\left[a\right]⊝\left[b\right]=\left[0\right]$ in ${\mathrm{\mathbb{Z}}}_n$, then neitherrole="math" localid="1646346676830" $\left[a\right]=\left[0\right]$ or $\left[b\right]=\left[0\right]$.

Step-by-step solution

Definition of multiplication ℤn 

According to the definition of multiplication in ${\mathrm{\mathbb{Z}}}_n$, we can say that $\left[a\right]⊝\left[c\right]=\left[ac\right]$.

Prove or disprove: If a⊝b=0 in ℤn, then a=0 or b=0.

Let us prove or disprove the result by considering the example.

For example, assume that$\left[2\right]$and$\left[3\right]$ belong in ${\mathrm{\mathbb{Z}}}_6$. Then find$\left[2\right]⊝\left[3\right]$ asfollows:

role="math" localid="1646346945193" $\begin{array}{rcl}\left[2\right]⊝\left[3\right]& =& \left[2·3\right]\\ & =& \left[6\right]\\ & =& \left[0\right]\end{array}$

This implies that$\left[2\right]⊝\left[3\right]$ is$\left[0\right]$ in ${\mathrm{\mathbb{Z}}}_6$.

We can observe that neither$\left[2\right]=\left[0\right]$ or $\left[3\right]=\left[0\right]$. Then the given result is false.

Hence,it is proved that when$\left[a\right]⊝\left[b\right]=\left[0\right]$ in ${\mathrm{\mathbb{Z}}}_n$, then neither$\left[a\right]=\left[0\right]$ or $\left[b\right]=\left[0\right]$.