Tazmin Horton

2021-04-04

The value of the operation [12]+[25] in ${Z}_{7}$ and to write the answer in the form [r] with $0\le r.

Cullen

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 $x=y\phantom{\rule{0.444em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}m\right)$. 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.
[12] + [25] = [12 + 25] = [37]
Since $37\equiv 2\phantom{\rule{0.444em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}7\right)$. Thus [12] + [25] = [2] in ${Z}_{7}$.
Therefore, the value of the expression is [2].

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