## BDMO Secondary National 2021 #7

Mehrab4226
Posts:230
Joined:Sat Jan 11, 2020 1:38 pm
BDMO Secondary National 2021 #7
কোনো ধনাত্মক পূর্ণসংখ্যা $$n$$-এর জন্য $$s(n)$$ আর $$c(n)$$ হলো যথাক্রমে $$n$$-এর পূর্ণবর্গ আর পূর্ণঘন উৎপাদকের সংখ্যা। একটা ধনাত্মক পূর্ণসংখ্যা $$n$$-কে নায্য বলা হবে যদি $$s(n)=c(n)>1$$ হয়। $$80$$-এর চেয়ে ছোট কতগুলো নায্য সংখ্যা আছে?

For a positive integer $n$ , let $s(n)$ and $c(n)$ be the number of divisors of $n$ that are perfect squares and perfect cubes respectively. A positive integer $n$ is called fair if $s(n)=c(n)>1$ . Find the number of fair integers less than $80$.
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

Pro_GRMR
Posts:46
Joined:Wed Feb 03, 2021 1:58 pm

### Re: BDMO Secondary National 2021 #7

Mehrab4226 wrote:
Sat Apr 10, 2021 1:26 pm
For a positive integer $n$ , let $s(n)$ and $c(n)$ be the number of divisors of $n$ that are perfect squares and perfect cubes respectively. A positive integer $n$ is called fair if $s(n)=c(n)>1$ . Find the number of fair integers less than $80$.
Let $n$ be in prime factorized form $n= p_1^{e_1}p_2^{e_2}\dots p_n^{e_n}$
We note that number of divisors of $p_1^{e_1}$ that are perfect square is $\lfloor{\frac{e_1}{2}}\rfloor+1$ similarly $\lfloor{\frac{e_1}{3}}\rfloor+1$ for perfect cubes.
So, $n$ has a total of $(\lfloor{\frac{e_1}{2}}\rfloor+1)(\lfloor{\frac{e_2}{2}}\rfloor+1)\dots(\lfloor{\frac{e_n}{2}}\rfloor+1)$ perfect square divisors and $(\lfloor{\frac{e_1}{3}}\rfloor+1)(\lfloor{\frac{e_2}{3}}\rfloor+1)\dots(\lfloor{\frac{e_n}{3}}\rfloor+1)$ perfect cube divisors.
So, for a number to be fair, $\lfloor \frac{e_i}{2} \rfloor = \lfloor \frac{e_i}{3} \rfloor$, which means that $e_i$ is either $1$ or $3$. So, a positive number $n$ is fair if and only if all its powers in the prime factorization are $1$ or $3$.
Now, we count the number of these numbers below $80$ and get all possible values of $n$ are $8, 24, 27, 40, 54, 56$ and so, $\boxed{6}$ values in total.
"When you change the way you look at things, the things you look at change." - Max Planck