Competition problem — Number theory

Library / /13 of 377

Number theory Difficulty 4.1 AIME Find the answer United States

Problem:

How many 2-digit positive integers have an even number of positive divisors?

Your answer

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

Solution

Solution:

An integer has an odd number of divisors precisely if it is a square. So we take the 90 2-digit numbers $10, 11, \ldots, 99$ and remove the 6 squares $4^{2}, 5^{2}, \ldots, 9^{2}$ , for a total of 84.

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