2025/26 AHSMC Part 1 Problem 10

Problem 10 of 16

Find the number of pairs $(a,b)$ of integers with $1 < a,b \le 30$ such that $\log_a b$ is an integer.

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Solution

Write $\log_a b = k$ with $k$ a positive integer, so $b = a^k$ and $1 < a \le 30$ , $1 < b \le 30$ .

If $k=1$ then $b=a$ , giving $29$ pairs ( $a=2,\ldots,30$ ).

If $k=2$ then $a^2 \le 30$ , so $a \in \{2,3,4,5\}$ : $4$ pairs.

If $k \ge 3$ then $a^3 \le 30$ , so $a \in \{2,3\}$ . For $a=2$ one has $3 \le k \le 4$ ; for $a=3$ one has $k=3$ . This is $3$ pairs.

Total: $29+4+3 = 36$ . The answer is (D).