Difference between revisions of "2023 IOQM/Problem 2"

Revision as of 20:36, 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, $x$ + $\frac{5}{x}$ =6. Upon solving the equation

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=54 pairs

@@ Line 6: / Line 6: @@
 ==Solution quick==
-<math>\log_{a}{b}=x</math>
+<math>\log_{a}{b}=x</math> Then, <math>x</math>+<math>\frac{5}{x}</math> =6. Upon solving the equation
-Then,
-x+5/x=6
-x^2-6x=5
 So, x equals to 2 or 3