Question

# The value of the operation [9]+[8] in Z_{13} and to write the answer in the form [r] with 0\leq r < m.

Congruence

The value of the operation $$[9]+[8] \in$$ $$\displaystyle{Z}_{{{13}}}$$ and to write the answer in the form [r] with $$\displaystyle{0}\leq{r}{<}{m}$$.

2021-03-24

Definition used:
Congruence.
Let m be an integer greater than 1. If x and y are integers, then x is congruent to y modulo m if $$x - y$$ is divisible by m. It can be represented as $$\displaystyle{x}={y}\pm{o}{d}{m}$$. This relation is called as congruence modulo m.
Calculation:
It is known that, $$[x] + [y] = [x+y]$$.
Then, the value of the expression becomes as follows.
$$[9] + [8] = [9 + 8]=[17]$$
Since $$\displaystyle{17}\equiv{4}\pm{o}{d}{13}$$. Thus $$[9] + [8] = [4] \in$$ $$\displaystyle{Z}_{{{13}}}$$.
Therefore, the value of the expression is [4].