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# In the froup Z_12, find |a|, |b|, and |a+b| a=6, b=2

Question
Abstract algebra
asked 2021-01-15
In the froup $$\displaystyle{Z}_{{12}}$$, find |a|, |b|, and |a+b|
a=6, b=2

## Answers (1)

2021-01-16
Consider the provided qustion,
If a $$\displaystyle\in{Z}_{{n}}$$ then $$\displaystyle{\left|{a}\right|}=\frac{{n}}{{{\gcd{{\left({a},{n}\right)}}}}}$$
here given $$\displaystyle{Z}_{{12}}$$. So, n=12
Given a = 6, b = 2
So, a + b = 6 + 2 = 8
$$\displaystyle{\left|{a}\right|}={\left|{6}\right|}=\frac{{12}}{{{\gcd{{\left({6},{12}\right)}}}}}=\frac{{12}}{{6}}={2}$$
$$\displaystyle{\left|{b}\right|}={\left|{2}\right|}=\frac{{12}}{{{\gcd{{\left({2},{12}\right)}}}}}=\frac{{12}}{{2}}={6}$$
$$\displaystyle{\left|{a}+\right|}={\left|{8}\right|}=\frac{{12}}{{{\gcd{{\left({8},{12}\right)}}}}}=\frac{{12}}{{4}}={3}$$

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