Sequential Fractions

Relevant wiki: Euler's Totient Function

We note that the general form of the fraction is $\dfrac n{2004-n} = \dfrac 1{\frac {2004}n - 1}$ , where $n =1,2,3,...2003$ . We also note that the fraction is irreducible when $\dfrac {2004}n$ is irreducible or $n$ and $2004$ are coprime integers. The number of 2003 fractions which is irreducible is the number of coprime integers of 2004 less than 2004, which is Euler's totient function of 2004:

$\begin{aligned} \phi (2004) & = 2004\left(1-\frac 1{\color{#3D99F6}{2}} \right)\left(1- \frac 1{\color{#3D99F6}{3}} \right) \left(1- \frac 1{\color{#3D99F6}{167}} \right) & \small \color{#3D99F6}{\text{where 2, 3 and 167 are prime factors of 2004.}} \\ & = 664 \end{aligned}$ .

Since: $\left \{\frac{1}{2003}, \frac{2}{2002}, \cdots \frac{1001}{1003}, \color{#3D99F6}{\frac{1002}{1002}}, \frac{1003}{1001}, \cdots \frac{2002}{2}, \frac{2003}{1} \right \} = \left \{\frac{1}{2003}, \frac{2}{2002}, \cdots \frac{1001}{1003}, \color{#3D99F6}{1}, \frac 1{\frac{1001}{1003}}, \cdots \frac 1{\frac 2{2002}}, \frac 1{\frac 1{2003}} \right \}$ , we note that half of the irreducible fractions are less than 1 and the other half greater than 1. Therefore, the number of irreducible less than 1 is $\dfrac {664}2 = \boxed{332}$ .

Hi Firstly, This question is asking for fractions lesser than one. Hence we only consider $\frac{1}{2003}$ to $\frac{1001}{1003}$ Now, for the fraction $\frac{n}{2004-n}$ to be reducible, n must be a factor or 2004. The factors of 2004 which are smaller than 1002 are 2,3.4.6,12,167,334,501,668(not considering 1). So what we are trying to find now is the number of fractions from $\frac{1}{2003}$ to $\frac{1001}{1003}$ which have their numerators being a factor of 2004. Hence we need to find the amount of numbers which are a multiple of 2,3,167 from 1-1001(The other factors are taken out as they are just multiple of those prime factors). Set A:Multiple of 2s Set B:Multiple of 3s Set C:Multiple of 167s AUBUC=A+B+C-AnB-BnC-AnC+AnBnC =500+333+5-166-1-2+0(not applicable) =669 This are the number of fractions which are REDUCIBLE. Hence the number of non-reducible fractions are 1001-669 =332 U stands for union while n stands for intersects(i am bad at latex)