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
ANSWERED
asked 2021-04-09

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

Answers (1)

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].

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