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

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

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Pohanginah

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