Maths Olympiad Prep

Track / Stage 5 / 127 of 400 #727 of 1964

Problem 727

AIME late
Number theory Difficulty 5.3 Find the answer

13. The greatest of the common divisors of two non-zero natural numbers is called the greatest common divisor (GCD) of these two numbers. If the GCD of two non-zero natural numbers is 1, then these two numbers are called coprime or relatively prime. For example, 2 and 3 are coprime, 3 and 8 are coprime; 12 and 15 are not coprime, because their GCD is 3. Among the natural numbers not exceeding 81, there are numbers that are coprime with 81.

A number or a short expression. Spacing, $ signs and \frac vs / are all fine.

Official solution

Analysis:
81=34,8181=3^{4}, 81 has only the prime factor 3, so all numbers containing the factor 3 are not coprime with 81, 81÷3=2781 \div 3=27, there are 27 numbers not greater than 81 that contain the factor 3, the remaining numbers are coprime with 81, 8127=5481-27=54 numbers.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.