Question

A multiplicative inverse of 3 modulo 5 is any integer a such that 3 a ≡ 1 (mod 5) , hence for some a ∈ Z5 .

Do such inverses exist for each element of Z5? If not, which ones?

Answer 1

the obvious element for which it can't exist is 0 as a*0=0 independent of modulo
all other elements have an inverse:
1*1≡1
2*3≡6≡1
3*2≡6≡1
4*4≡16≡1

if there are more than a few numbers/guessing is inefficient it can be calculated using the extended euclidean algorithm