2023 IOQM/Problem 2
Problem
Find the number of elements in the set
![\[\lbrace(a.b)\in N: 2 \leq a,b \leq2023,\:\: \log_{a}{b}+6\log_{b}{a}=5\rbrace\]](http://latex.artofproblemsolving.com/7/5/f/75fe09afd4da734f976243ddeda521b906b7a04b.png)
Solution quick
$$ (Error compiling LaTeX. Unknown error_msg)assume \log_{a}{b}=x$$ Then, \begin{align*} x + 5/x &= 6 & \text{because log} \\ x^2 + 5 &= 6x & \text{by such-and-such} \end{align*}
So, x equals to 2 or 3
For 2, it's obvious that there are {2,4}...{44,1936}
For 3, it's obvious that there are {2,8}...{12,1728}
Thus, there are 43+11=53 pairs
