Multiplicative group of integers modulo m Modular multiplicative inverse

not every element of complete residue system modulo m has modular multiplicative inverse, instance, 0 never does. after removing elements of complete residue system not relatively prime m, left called reduced residue system, of elements have modular multiplicative inverses. number of elements in reduced residue system



ϕ
(
m
)


{\displaystyle \phi (m)}

,



ϕ


{\displaystyle \phi }

euler totient function, i.e., number of positive integers less m relatively prime m.

in general ring unity not every element has multiplicative inverse , called units. product of 2 units unit, units of ring form group, group of units of ring , denoted r if r name of ring. group of units of ring of integers modulo m called multiplicative group of integers modulo m, , isomorphic reduced residue system. in particular, has order (size),



ϕ
(
m
)


{\displaystyle \phi (m)}

.

in case m prime, p,



ϕ
(
p
)
=
p
−
1


{\displaystyle \phi (p)=p-1}

, non-zero elements of




z


/

p

z



{\displaystyle \mathbb {z} /p\mathbb {z} }

have multiplicative inverses,




z


/

p

z



{\displaystyle \mathbb {z} /p\mathbb {z} }

finite field. in case, multiplicative group of integers modulo p form cyclic group of order p − 1.