Which class in $\mathbb{Z}_{10}$ has no multiplicative inverse?

Explanation

1. 01

   $[a]$ has an inverse modulo 10 exactly when $a$ and 10 have no common factor.

2. 02

   4 shares the factor 2 with 10, so $4x$ is always even and never $\equiv 1$. The others: $3 \times 7 = 21$, $7 \times 3 = 21$, $9 \times 9 = 81$, all $\equiv 1$.