BdMO National Higher Secondary 2020 P2

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BdMO National Higher Secondary 2020 P2

Unread post by Mursalin » Mon Feb 08, 2021 4:08 pm

কতগুলো পূর্ণসংখ্যা \(n\) আছে যেন \(1\leq n\leq 2020\) এবং \(n^n\) একটা পূর্ণবর্গ সংখ্যা?

How many integers \(n\) are there subject to the constraint that \(1\leq n\leq 2020\) and \(n^n\) is a perfect square?
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Re: BdMO National Higher Secondary 2020 P2

Unread post by Mehrab4226 » Tue Feb 09, 2021 12:15 am

It can easily be seen that all even $n$ has the property. And also all the odd perfect square $n$ possible. That would make $1010+22 = 1032$ ($22$ because $45^2=2025 >2020)$

Other values of $n$ won't work because in their prime power factorization there is a prime with an odd-numbered power and when we power odd n over n then the power of that prime still remains odd. Thus the number can never be a square
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

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