20!=2^{\alpha_{1}}\cdot 3^{\alpha_{2}}\cdot 5^{\alpha_{3}}\cdot 7^{\alpha_{4}}\cdot 11^{\alpha_{5}}\cdot 13^{\alpha_{6}}\cdot 17^{\alpha_{7}}\cdot 19^{\alpha_{8}}\\

\alpha_{1}=\left[\frac{20}{2}\right]+\left[\frac{20}{4}\right]+\left[\frac{20}{8}\right]+\left[\frac{20}{16}\right]=10+5+2+1=18\\

\alpha_{2}=\left[\frac{20}{3}\right]+\left[\frac{20}{9}\right]=6+2=8\\

\alpha_{3}=\left[\frac{20}{5}\right]=4\\

\alpha_{4}=\left[\frac{20}{7}\right]=2\\

\alpha_{5}=\left[\frac{20}{11}\right]=1\\

\alpha_{6}=\left[\frac{20}{13}\right]=1\\

\alpha_{7}=\left[\frac{20}{17}\right]=1\\

\alpha_{8}=\left[\frac{20}{19}\right]=1\\