Given a number N. The task is to find the number of unordered coprime pairs of integers from 1 to N. There can be multiple queries.
Input: 3 Output: 4 (1, 1), (1, 2), (1, 3), (2, 3) Input: 4 Output: 6 (1, 1), (1, 2), (1, 3), (1, 4), (2, 3), (3, 4)
Approach: Here Euler’s Totient Function will be helpful. Euler’s totient function denoted as phi(N), is an arithmetic function that counts the positive integers less than or equal to N that are relatively prime to N.
The idea is to use the following properties of Euler Totient function i.e.
- The formula basically says that the value of Φ(n) is equal to n multiplied by product of (1 – 1/p) for all prime factors p of n. For example value of Φ(6) = 6 * (1-1/2) * (1 – 1/3) = 2.
- For a prime number p, Φ(p) is p-1. For example Φ(5) is 4, Φ(7) is 6 and Φ(13) is 12. This is obvious, gcd of all numbers from 1 to p-1 will be 1 because p is a prime.
Now, find the sum of all phi(x) for all i between 1 to N using prefix sum method. Using this, one can answer in o(1) time.
Below is the implementation of above approach.
Number of unordered coprime pairs of integers from 1 to 3 are 4 Number of unordered coprime pairs of integers from 1 to 4 are 6
Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.
- Count all pairs of divisors of a number N whose sum is coprime with N
- Count unordered pairs (i,j) such that product of a[i] and a[j] is power of two
- Count of unordered pairs (x, y) of Array which satisfy given equation
- Check if all the pairs of an array are coprime with each other
- XOR of pairwise sum of every unordered pairs in an array
- Bitwise XOR of all unordered pairs from a given array
- Bitwise OR of all unordered pairs from a given array
- Count of integers up to N which are non divisors and non coprime with N
- Largest number less than or equal to N/2 which is coprime to N
- Coprime divisors of a number
- Finding a Non Transitive Coprime Triplet in a Range
- Composite XOR and Coprime AND
- Length of the longest increasing subsequence such that no two adjacent elements are coprime
- Largest Coprime Set Between two integers
- Find the length of the Largest subset such that all elements are Pairwise Coprime
- Print all distinct Coprime sets possible from 1 to N
- Print all Coprime path of a Binary Tree
- Construct an array of first N natural numbers such that every adjacent pair is coprime
- Find the missing number in unordered Arithmetic Progression
- Find sum of xor of all unordered triplets of the array
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to email@example.com. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.