Question

Find the sum of all positive rational numbers that are less than $10$ and that have denominator $30$ when written in lowest terms.

Answer

Answer: 400

Concepts:Euler’s Totient Function fraction summation

Difficulty rating: 1780

Small Hint:

Write every number as $\frac{k}{30}$ and impose the condition $\gcd(k,30)=1$

Big Hint:

Group the eligible numerators into ten blocks of length $30$

Solution:

The numbers are $\frac{k}{30}$ for $1\leq k\lt300$ and $\gcd(k,30)=1.$ There are $\varphi(30)=8$ eligible residues in each block of $30,$ and their sum is $\frac{30\varphi(30)}{2}=120.$ Thus the sum of all eligible numerators is $\begin{aligned}\sum_{q=0}^9(8\cdot30q+120)&=8\cdot30\cdot45\\&\quad+10\cdot120\\&=12000.\end{aligned}$ Dividing by $30$ gives $400.$

1992 AIME Problem 1

Answer: 400

Difficulty rating: 1780

Small Hint:

Big Hint:

Solution:

Problem 1 in Other Years