For $t \ge 2$, define $S(t)$ as the number of times $t$ divides into $t!$. We say that a positive integer $t$ is a peak if $S(t) > S(u)$ for all values of $u < t$. Prove or disprove the following statement: For every prime $p$, there is an integer $k$ for which $p$ divides $k$ and $k$ is a peak.