$n$ is a number such as $n \in \mathbb{N}$ and $n >0$.(Eg. $n = 8$)
$p$ is the sum of the digits of $n$ in base $10$.(Eg. $n=80$, $a = 8+0 = 8$)
$q$ is the sum of the digits of $1/n$ in base $10$.(Eg. $n=80$, $1/n = 0.0125$, $b = 0+0+1+2+5 = 8$)
If $p = q$, the number is called a cool number.
If $p \neq q$, the number is called a not cool number.
The question is : which numbers $n$ are cool ?
What I found myself :
- Numbers with infinite decimals can't be cool.
- If $n$ is cool, then $n = 2^a*5^b$ with $a \in \mathbb{N}$ and $b \in \mathbb{N}$
- If $ 2^a*5^b$ is cool, then $2^{a+1}*5^{b+1}$ is cool too.
- If $ 2^a*5^b$ is not cool, then $2^{a+1}*5^{b+1}$ is not cool too.
- These numbers are cool : $1,8,10,80,100,125,800,1000,1250,8000...$
Thank you for helping in my quest for cool numbers.