The last non-zero digit in $20!$ is

 $20!=2^{\alpha }{.3}^{\beta }{.5}^r{.7}^{\alpha 1}{.11}^{\alpha 2}{.13}^{\alpha 3}{.17}^{\alpha 4}{.19}^{\alpha 5}$
 $20!=2^{18}{.3}^8{.5}^4{.7}^2{.11}^1{.13}^1\mathrm{.17.19}=2^{14}{.3}^8{.7}^2{.11}^1\mathrm{.13.17.19}$
 $$\begin{gathered} =\left(\text{ends with 4}\right)\left(\text{ends with 1}\right)\left(\text{ends with 9}\right)\left(\text{ends with 1}\right)\left(\text{ends with 3}\right) \\ \left(\text{ends with 7}\right)\left(\text{ends with 9}\right) \end{gathered}$$
 $108\times 63=\text{ ends with 4}$