Difference between revisions of "2023 IOQM/Problem 2"

Revision as of 09:59, 2 October 2023

Find the number of elements in the set

$\lbrace(a.b)\in N: 2 \leq a,b \leq2023,\:\: \log_{a}{b}+6\log_{b}{a}=5\rbrace$

$\log_{a}{b}=x$ Then,

x+5/x=6

x^2-6x=5

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

@@ Line 6: / Line 6: @@
 ==Solution quick==
-<math></math>assume \log_{a}{b}=x\$$
+<math>\log_{a}{b}=x</math>
 Then,
-\begin{align*}
-x + 5/x &= 6 & \text{because log} \\
+x+5/x=6
-x^2 + 5 &= 6x & \text{by such-and-such}
-\end{align*}
+x^2-6x=5
 So, x equals to 2 or 3