In the froup Z_12, find |a|, |b|, and |a+b| a=3, b=8

Question
Abstract algebra
asked 2020-12-30
In the froup \(\displaystyle{Z}_{{12}}\), find |a|, |b|, and |a+b|
a=3, b=8

Answers (1)

2020-12-31
Consider the provided qustion,
If a \(\displaystyle\in{Z}_{{n}}\) then |a|=n/(gcd(a,n))ZSK
here given \(\displaystyle{Z}_{{12}}\). So, n=12
Given a = 6, b = 2
So, a + b = 6 + 2 = 8
\(\displaystyle{\left|{a}\right|}={\left|{3}\right|}=\frac{{12}}{{{\gcd{{\left({3},{12}\right)}}}}}=\frac{{12}}{{3}}={4}\)
\(\displaystyle{\left|{b}\right|}={\left|{8}\right|}=\frac{{12}}{{{\gcd{{\left({8},{12}\right)}}}}}=\frac{{12}}{{4}}={3}\)
\(\displaystyle{\left|{a}+{b}\right|}={\left|{11}\right|}=\frac{{12}}{{{\gcd{{\left({11},{12}\right)}}}}}=\frac{{12}}{{1}}={12}\)
0

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