Question 55
Given natural number $a \gt 1$ and different odd prime numbers $p_1, \ldots, p_n$, with
$$
a^{p_1} \equiv 1 \pmod{p_2},\ a^{p_2} \equiv 1 \pmod{p_3},\ \ldots,\ a^{p_n} \equiv 1 \pmod{p_1}.
$$
Prove that
a) $(a-1)$ is divisible by $p_i$ for some $i = 1, \ldots, n$.
b) Can $(a-1)$ be divisible by $p_i$ for exactly one $i$ of $i = 1, \ldots, n$?
$$
a^{p_1} \equiv 1 \pmod{p_2},\ a^{p_2} \equiv 1 \pmod{p_3},\ \ldots,\ a^{p_n} \equiv 1 \pmod{p_1}.
$$
Prove that
a) $(a-1)$ is divisible by $p_i$ for some $i = 1, \ldots, n$.
b) Can $(a-1)$ be divisible by $p_i$ for exactly one $i$ of $i = 1, \ldots, n$?