My question is derived from BMO 1984 question 4.
Given an integer n, how many r (with r bigger than 0 less than 1) can make 2nr an integer?
I tried some values of n and from 1 to 9 there are 1,2,3,3,3,5,3,4,5 r that meet the requirement.
I find it hard to generalise it to n.(Maybe I have done something wrong?)
Could anyone help me please? Thanks.
First observe that the condition $0<r<1$ implies $0<2nr<2n$. Next observe that for any integer $m, 0 < m< 2n$, there is precisely one $r\in(0,1)$ with $m=2nr$ (namely, $r=m/2n$). Therefore the question is equivalent to the following question: for an integer $n$, how many integers $m$ satisfy $0 < m < 2n$. Clearly this is $2n-1$.