Question

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

Abstract algebra
ANSWERED
asked 2021-02-27

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

Answers (1)

2021-02-28

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, \(a + b = 5 + 4 = 9\)
\(\displaystyle{\left|{a}\right|}={\left|{5}\right|}=\frac{{12}}{{{\gcd{{\left({5},{12}\right)}}}}}=\frac{{12}}{{1}}={12}\)
\(\displaystyle{\left|{b}\right|}={\left|{4}\right|}=\frac{{12}}{{{\gcd{{\left({4},{12}\right)}}}}}=\frac{{12}}{{4}}={3}\)
\(\displaystyle{\left|{a}+{b}\right|}={\left|{11}\right|}=\frac{{12}}{{{\gcd{{\left({9},{12}\right)}}}}}=\frac{{12}}{{3}}={4}\)

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