Question

# The value of the operation [13]^{6} in Z_{15} and to write the answer in the form [r] with 0\leq r < m.

Polynomial factorization

The value of the operation $$\displaystyle{\left[{13}\right]}^{{{6}}}$$ in $$\displaystyle{Z}_{{{15}}}$$ and to write the answer in the form [r] with $$\displaystyle{0}\leq{r}{<}{m}$$.

2021-04-11

Definition used:
Congruence.
Let m be an integer greater than 1. If xand yare integers, then x is congruent to y modulo m if x — y is divisible by m. It can be represented as $$\displaystyle{x}={y}{\left({b}\ mod\ {m}\right)}$$. This relation is called as congruence modulo m.
Calculation:
Consider the values and congruence classes modulo m,
$$\displaystyle{\left[{13}\right]}^{{{6}}}$$ and m = 15.
It is evident that [x]=[y] if $$\displaystyle{\left[{x}={y}{\left({b}\ mod\ {m}\right)}\right]}$$. Since $$\displaystyle{13}=-{2}{\left({b} \ mod \ {15}\right)}$$.
Then,
[13]=[-2]
$$\displaystyle{\left[{13}\right]}^{{{6}}}={\left[-{2}\right]}^{{{6}}}$$
$$\displaystyle={\left[{\left(-{2}\right)}^{{{6}}}\right]}$$
=[64]
Since, by the principal or congruence $$\displaystyle{64}={4}{\left({b}\ mod \ {15}\right)}$$. Therefore, $$\displaystyle{\left[{13}\right]}^{{{6}}}={\left[{4}\right]}$$ in $$\displaystyle{Z}_{{{15}}}$$.
Therefore, the value of the expression is [4].