It can also be shown that for arbitrary positive integer \(a, \phi(a)\) is given by

\[\phi(a)=\prod_{i=1}^{t}\left[p_{i}^{a_{i}-1}\left(p_{i}-1ight)ight]\]

where \(a\) is given by Equation (8.1), namely: \(a=P_{1}^{a_{1}} P_{2}^{a_{2}} \ldots P_{t}^{a_{t}}\). Demonstrate this result.