A Question?

Question

if p is a prime, please explain why all elements in Z/pZ have multiplicative inverse?

where Z is integers

lolita what is cero in the first line?

Answer 1

suppose that there exist an a not cero, such that there is a b different form cero such that:
ab=0 mod p
that means that
ab=pn,
but since p is a prime, it would mean that
p divides a
or p divides b,
but this cannot happen since 0<=a,b then the only posibility is for n=0,
but then ab=0 (in the integers) which means that
a =0 or b=0, which contradicts the hypothesis at the begining

notice that the fact that p is a prime is strongly used,
for example
if you consider Z/4Z,
we have
2x2=4, but 4 is not a prime, so we cannot say that 4 has to divide 2...(not at all.!)