Difference between revisions of "2023 IOQM/Problem 2"

Revision as of 21:32, 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=54 pairs

Revision as of 10:59, 2 October 2023 (view source) Andy666 (talk \| contribs) (→‎Solution quick) ← Older edit		Revision as of 21:32, 2 October 2023 (view source) Sansgankrsngupta (talk \| contribs) (→‎Solution quick) Newer edit →
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	For 3, it's obvious that there are {2,8}...{12,1728}		For 3, it's obvious that there are {2,8}...{12,1728}

−	Thus, there are 43+11=53 pairs	+	Thus, there are 43+11=54 pairs