How many positive fractions in simplest form have denominators of 100 where the numerator does not share a common factor other than 1 with the denominator 100?

Answer 1

There are 40.

The question is asking how how many numbers 1-99 are relatively prime to 100.

The prime factorisation of 100 is 22 x 52, so all numbers 1-99 which do not have either of 2 and 5 in their prime factorization will have no common factor with 100 other than 1. ie the numbers which are some of the primes 3, 7, 11, 13, ... multiplied together, along with the unit fraction (1/100).

Thus there are:

• The 1 unit fraction
• The 23 primes {3, 7, ..., 97};
• The 9 composites of 3 with another one of the primes {3, 7, ..., 31};
• The 3 composites of 32 with another one of the primes {3, 7, 11};
• The 1 composite of 34;
• The 3 composites of 7 with one other prime factor {7, 11, 13}.

Making 1 + 23 + 9 + 3 + 1 + 3 = 40 positive fractions in simplest form with a denominator of 100 and no common factor other than 1.