Putnam 1984 A6

Let $n$ be a positive integer, and let $f(n)$ denote the last nonzero digit in the decimal expansion of $n!$ . $(\text a)$ Show that if $a_1,a_2,\ldots,a_k$ are distinct nonnegative integers, then $f(5^{a_1}+5^{a_2}+\ldots+5^{a_k})$ depends only on the sum $a_1+a_2+\ldots+a_k$ . $(\text b)$ Assuming part $(\text a)$ , we can define $$g(s)=f(5^{a_1}+5^{a_2}+\ldots+5^{a_k}),$$ where $s=a_1+a_2+\ldots+a_k$ . Find the least positive integer $p$ for which $$g(s)=g(s+p),\enspace\text{for all }s\ge1,$$ or show that no such $p$ exists.