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Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Basics of Mathematics

Olympiad workbook · IOQM · Basics of Mathematics

Find the number of elements in the set \ (a, b) ∈ N: 2 ≤ a, b ≤...

Find the number of elements in the set {(a,b)∈N:2≤a,b≤2023,log⁡a(b)+6log⁡b(a)=5}\left\{(a, b) \in N: 2 \leq a, b \leq 2023, \log _a(b)+6 \log _b(a)=5\right\}.

Correct answer

54

Step-by-step solution

log⁡ab=tt+6t=5⇒t2−5t+6=0⇒t=2 or 3log⁡ab=2 or 3⇒b=a2 or a3a2≤2023 and a3≤2023⇒a≤44.97 and a≤12.64⇒a∈{2,….44} and a∈{2,….12}⇒(44−2+1)+(12−2+1)⇒43+11=54\begin{aligned} & \log _a b=t \\ & t+\frac{6}{t}=5 \\ & \Rightarrow t^2-5 t+6=0 \quad \Rightarrow t=2 \text { or } 3 \\ & \log _a b=2 \text { or } 3 \Rightarrow b=a^2 \text { or } a^3 \\ & a^2 \leq 2023 \text { and } a^3 \leq 2023 \\ & \Rightarrow \quad a \leq 44.97 \text { and } a \leq 12.64 \\ & \Rightarrow \quad a \in\{2, \ldots .44\} \text { and } a \in\{2, \ldots .12\} \\ & \Rightarrow \quad(44-2+1)+(12-2+1) \\ & \Rightarrow \quad 43+11=54\end{aligned}
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