Difference between revisions of "2023 IOQM/Problem 2"

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[[IOQM]]
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'''Please note that all problems on this page are copyrighted by THE [https://www.mtai.org.in | MTA(I)]'''

Revision as of 06:16, 5 October 2023

Problem

Find the number of elements in the set

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

Solution1(Quick)

Finding the no. of elements in the set means finding no. of ordered pairs of ($a$, $b$)

$\log_{a}{b}=x$ Then, $\log_{b}{a}=\frac{1}{x}$.

$\implies$ $x$+$\frac{6}{x}$ =5. Upon simplifying, we get $x^{2}-5x+6=0$

$\implies$ $(x-2)(x-3)=0$

So, $x$ equals to 2 or 3

For $x$ = 2, it implies that $\log_{a}{b}=2$. So, $a\:=\: b^{2}$, Hence all such pairs are of the form ($b^{2}$,$b$)

Where each number lies between 2 and 2023 (inclusive). All such pairs are (4, 2);(9, 3);(16, 4);........(1936, 44)

Total no. of these pairs = 43

For $x$ = 3, Following a similar pattern, we get the pairs as {2,8}...{12,1728} ($b^{3}$,$b$)

Total no. of these pairs = 11

Thus, there are 43+11=$\boxed{54}$ elements in the set

~ SANSGANKRSNGUPTA AND ~Andy666

Video Solutions

Video solution by cheetna: https://www.youtube.com/watch?v=z0NPa0tUzZk

Video solution by Unacademy Olympiad Corner: https://www.youtube.com/watch?v=Mm6mXjwU9bY

Video solution by Vedantu Olympiad School: https://www.youtube.com/watch?v=4DJXtR4VHEA

Video solution by Olympiad Wallah: https://www.youtube.com/watch?v=4HSjmY7d3nA

Video solution by : Motion Olympiad Foundation Class 5th - 10th: https://www.youtube.com/watch?v=oVaeHceHXsQ

Please note that above videos solutions are in Hindi, some in English and some in mixed(Hindi + English).

~SANSGANKRSNGUPTA

See Also

IOQM

Mathematics competitions

Please note that all problems on this page are copyrighted by THE | MTA(I)