BdMO National 2021 Junior Problem 9

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Anindya Biswas
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BdMO National 2021 Junior Problem 9

Unread post by Anindya Biswas » Mon Apr 12, 2021 12:14 pm

এক জোড়া অসমান পূর্ণসংখ্যাকে বন্ধুসুলভ বলা হবে যদি তারা পরস্পর সহমৌলিক না হয়। \(1\), \(2\), \(3\), \(4\), \(5\), \(6\), \(7\), \(8\) সংখ্যাগুলো দিয়ে সর্বোচ্চ দুটো নিশ্ছেদ বন্ধুসুলভ জোড়া বানানো সম্ভব। যেমন \((2, 4)\) আর \((3, 6)\)। \(1, 2, 3\cdots, 50\) সংখ্যাগুলো দিয়ে কতগুলো নিশ্ছেদ বন্ধুসুলভ জোড়া বানানো সম্ভব?

A pair of distinct integers are called friendly if they are not coprime. Using the numbers $1,2,3,4,5,6,7,8$, at most $2$ disjoint friendly pairs can be formed, for example: $(2,4)$ and $(3,6)$. How many disjoint friendly pairs can be formed using the numbers $1,2,3,\cdots,50$?
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

Naeem588
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Joined:Sat Apr 03, 2021 1:41 am

Re: BdMO National 2021 Junior Problem 9

Unread post by Naeem588 » Wed Aug 18, 2021 11:53 am

We can only use Composite numbers in our pairs. There are 34 composite numbers between 1-50. But the Square of 2,3,5&7 exist Between 1-50. So we can use Them also in our pairs. So we get a total of 38 numbers. So the highest number of pairs possible is 19.
Here is a example
(50,28), (49,7), (48,46), (44,42), (32,38), (36,34), (45,40), (39,33), (35,30), (27,24), (26,22), (25,5), (21,18), (20,16), (15,10), (14,12), (9,3), (8,6), (4,2).

Sah Alom
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Re: BdMO National 2021 Junior Problem 9

Unread post by Sah Alom » Sun Jan 16, 2022 4:12 pm

What was the motivation for this approach?

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