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 $1,2,3,\cdots,50$?
BdMO National 2021 Junior Problem 9
- Anindya Biswas
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"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
— John von Neumann
— John von Neumann
Re: BdMO National 2021 Junior Problem 9
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).
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).
Re: BdMO National 2021 Junior Problem 9
What was the motivation for this approach?