# In the froup Z_12, find |a|, |b|, and |a+b|a=5, b=4

In the froup $$\displaystyle{Z}_{{12}}$$, find $$|a|, |b|$$, and $$|a+b|$$
$$a=5, b=4$$

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Pohanginah

Consider the provided qustion,
If a in $$\displaystyle{Z}_{{n}}$$ then $$\displaystyle{\left|{a}\right|}=\frac{{n}}{{{\gcd{{\left({a},{n}\right)}}}}}$$
here given $$\displaystyle{Z}_{{12}}$$. So, $$n=12$$
Given $$a = 5, b = 4$$
So, $$a + b = 5 + 4 = 9$$
$$\displaystyle{\left|{a}\right|}={\left|{5}\right|}=\frac{{12}}{{{\gcd{{\left({5},{12}\right)}}}}}=\frac{{12}}{{1}}={12}$$
$$\displaystyle{\left|{b}\right|}={\left|{4}\right|}=\frac{{12}}{{{\gcd{{\left({4},{12}\right)}}}}}=\frac{{12}}{{4}}={3}$$
$$\displaystyle{\left|{a}+{b}\right|}={\left|{11}\right|}=\frac{{12}}{{{\gcd{{\left({9},{12}\right)}}}}}=\frac{{12}}{{3}}={4}$$