Question

If $p\ge 5$and $p$is prime, prove that$\left[p\right]=\left[1\right]$or $\left[p\right]=\left[5\right]$in ${\mathrm{\mathbb{Z}}}_6$.

Short answer

It is proved$\left[p\right]=\left[1\right]$or$\left[p\right]=\left[5\right]$ in${\mathrm{\mathbb{Z}}}_6$ .

Step-by-step solution

Consider the given situation

Let us consider the integers $a,b$and $n$which satisfy the relation $a\equiv b\left(modn\right)$. This means that there exists an integer $k$, which holds the property:

$a-b=nk$

Assume that p is a prime with $p\ge 5$.

As $p\ge 5$, the possible forms areas follows:

$p=6k,p=6k+1,p=6k+2,p=6k+4,p=6k+5$

Prove that  p=1or p=5 in ℤ6 

Let us consider$p=6k,p=6k+2$and$p=6k+4$ .This implies that p is an even number,but this contradictsthe fact that p is a prime number.

Therefore, $p=6k+1$and $p=6k+5$. This implies that

$p\equiv 1mod6$ and $p\equiv 5mod6$.

Then by theorem 2.3, that is $a\equiv c\left(modn\right)$, if and only if $\left[a\right]=\left[c\right]$. We have$\left[p\right]=\left[1\right]or\left[5\right]$ .

It is proved that $\left[p\right]=\left[1\right]$or$\left[p\right]=\left[5\right]$ in${\mathrm{\mathbb{Z}}}_6$ .