I don’t understand the formulation of the following problem:
We call a natural number $b$ "happy" if for every natural number $a$ such that $a^5$ is divisible by $b^2$ it follows that $a^2$ is divisible by $b$. How many "happy" numbers below $2018$ are there.
My confusion is, isn’t this true for all $a$, $b$?
N.B. do NOT give a solution, only help with the formulation.
Thank you
I think you are just overthinking the problem statement. Who cares if it works for any number that is a multiple of $b$? That will contribute to the count of the number of happy numbers below 2018, but that has nothing to do with the problem formulation.
Someone deleted a comment that I think is relevant, and that is to try a computational example:
Let $b=3$. Then if $a^5$ is divisible by $9$, $a^5$ is divisible by $3$, and thus $a$ is divisible by $3$, and thus $a^2$ is divisible by $3$. This makes $b$ happy.
However, if $b=3^5$, and $a=3^2$, then $a^5=3^{10}$ is divisible by $b^2=3^{10}$, but $a^2=3^4$ is not divisible by $b$.