HMMT November, 2013

Number theory Difficulty 4.5 AIME Find the answer United States

Problem:

How many positive integers less than $100$ are relatively prime to $200$ ? (Two numbers are relatively prime if their greatest common factor is $1$ .)

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

A positive integer $n < 100$ is relatively prime to $200$ if and only if it is relatively prime to $100$ (since $200$ and $100$ have the same prime factors). Thus, our answer is $\varphi(100) = 100 \times \frac{1}{2} \times \frac{4}{5} = 40$ .

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.