Let $n=2^km$ be a positive integer with $m$ odd and $r$ the largest prime that divides $m$. Then are there at least $\frac{r-1}{2}$ numbers coprime to $n$ which are less than $r$?
I think yes, but am unable to clearly see it. There are $r-1$ numbers coprime to $r$ less than $r$. But does this translate to half of them being coprime to $n$? What if I replace $r$ as the largest prime to the smallest prime that divides $n$? Any hints? Thanks beforehand.
This is easily seen to be false. Thanks to comment by @abc... consider the number $n=210=2\cdot3\cdot5\cdot7$. Then $r$ is $7$ but there are not $3$ numbers that are coprime to $210$ less than $7$