A Simple Problem From The 2024 IOQM

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A Simple Problem From The 2024 IOQM

This is a relatively simple problem from the 2024 Indian Olympiad Qualifier.

Problem contributed by debosmita1729

Let $\log_a(b) = t$

$t + \dfrac{6}{t} = 5$

$\Rightarrow t^2 - 5t + 6 = 0$

$\Rightarrow t = 2$ or $3$

$\Rightarrow \log_a(b) = 2$ or $3$

$\Rightarrow b = a^2$ or $a^3$

From the bounds given,

$a^2 \leq 2023$ and $a^3 \leq 2023$

$\Rightarrow a \in \{2,...,44\}$ and $a \in \{2,...,12\}$

So, the number possible values of $(a,b)$ is:

$(44 - 2 + 1) + (12 - 2 + 1) = 43 + 11 = 54$