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, \(a + b = 5 + 4 = 9\)

\(\displaystyle{\left|{a}\right|}={\left|{5}\right|}=\frac{{12}}{{{\gcd{{\left({5},{12}\right)}}}}}=\frac{{12}}{{1}}={12}\)

\(\displaystyle{\left|{b}\right|}={\left|{4}\right|}=\frac{{12}}{{{\gcd{{\left({4},{12}\right)}}}}}=\frac{{12}}{{4}}={3}\)

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