Competition problem — Number theory

Library / /32 of 348

Number theory Difficulty 4.6 AIME Find the answer

How many positive integers less than 100 are relatively prime to 200?

Your answer

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

Solution

$401 \leq n<100$ is relatively prime to 200 if and only if it's relatively prime to 100 (200, 100 have the same prime factors). Thus our answer is $\phi(100)=100 \frac{1}{2} \frac{4}{5}=40$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.