Question 13
Denote by $p(a)$ the first digit of the natural number $a$. Show that each of the sets
$$
A = \{n \in \mathbb{N} \mid p(5^n) - p(2^n) \gt 0\}, \quad B = \{n \in \mathbb{N} \mid p(5^n) - p(2^n) \lt 0\}
$$
has infinitely many elements.
$$
A = \{n \in \mathbb{N} \mid p(5^n) - p(2^n) \gt 0\}, \quad B = \{n \in \mathbb{N} \mid p(5^n) - p(2^n) \lt 0\}
$$
has infinitely many elements.