Question 1
Let $B_n$ be a number of ways to partition a $n$-element set onto non-empty parts. For example, $B_3 = 5$ because we have the following partitions of the 3-element set $\{a, b, c\}$:
$$
\{a, b, c\}; \quad \{a\}, \{b, c\}; \quad \{a, c\}, \{b\}; \quad \{a, b\}, \{c\}; \quad \{a\}, \{b\}, \{c\}.
$$
Prove that for every positive integer $m$ and prime number $p$,
$$
B_{p^m} \equiv m + 1 \pmod{p}.
$$
$$
\{a, b, c\}; \quad \{a\}, \{b, c\}; \quad \{a, c\}, \{b\}; \quad \{a, b\}, \{c\}; \quad \{a\}, \{b\}, \{c\}.
$$
Prove that for every positive integer $m$ and prime number $p$,
$$
B_{p^m} \equiv m + 1 \pmod{p}.
$$