21. A sequence of numbers is arranged according to the rule:
$$
1^{1}, 2^{2}, 3^{3}, 4^{4}, 5^{5}, 1^{6}, 2^{7}, 3^{8}, 4^{9}, 5^{10}, \cdots \cdots
$$

The bases cycle through $1 \sim 5$, and the exponents start from 1 and increase sequentially. Let the sum of the first $n$ numbers be denoted as $P(n)$. When $P(n)$ is a multiple of 10 for the first time, $n=$ $\qquad$ .
$37$