How many (distinct) numbers between 0 and 1 (inclusive) can be expressed as a fraction m/n , where m and n are integers such that m+n=200 ?
To find the number of distinct numbers between 0 and 1 that can be expressed as a fraction m/n, where m and n are integers such that m+n=200, we can use the concept of Farey sequences.
A Farey sequence of order n is a sequence of fractions that lie between 0 and 1, with denominators less than or equal to n, arranged in increasing order.
In this case, we have m + n = 200, which means the denominator of the fractions can be at most 200.
Now, let's calculate the number of terms in each Farey sequence of order n. The number of terms in a Farey sequence F(n) can be given as the sum of Euler's totient function (phi) values from 1 to n.
So, we need to calculate phi(1) + phi(2) + phi(3) + ... + phi(200) to find the number of terms in the Farey sequence F(200).
To find the phi function values, we can use a table or a formula. The phi function (phi) of a positive integer n represents the count of positive integers less than or equal to n that are relatively prime to n.
Once we have the phi values from 1 to 200, we can sum them up to get the total number of terms in the Farey sequence F(200).
This method gives us a way to calculate the number of distinct numbers between 0 and 1 that can be expressed as fractions with m and n as integers, where m + n = 200.