jmc

We are looking for an integer $a$ such that $4a$ is congruent to 1 modulo 21. One approach is to check integers of the form $21k+1$, where $k\ge 0$ is an integer, for divisibility by 4. We find that 22 and 43 are not divisible by 4, but $21(3)+1=64$ is equal to $4\times 16$. Thus $\boxed{16}$ times 4 is congruent to 1 modulo 21.