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

In the froup ${Z}_{12}$, find $|a|,|b|$, and $|a+b|$
$a=3,b=8$

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Talisha

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