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2023 IOQM/Problem 2

Revision as of 10:59, 2 October 2023 by Andy666 (talk | contribs) (→‎Solution quick)

Problem

Find the number of elements in the set

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

Solution quick

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

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