Given an integer N. The task is to count the number of ordered pairs (a, b) such that .
Input: N = 5 Output: 8 Ordered Pairs are = (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (3, 1), (4, 1). Input: N = 1000 Output: 7053
Naive Approach: Run two for loops upto N – 1 and count ordered pairs whose product are less than N.
Efficient Approach: Let’s considered an ordered pair (a, b). Then, if the product of two numbers is less than n i:e a * b < n, then at least one of them must be less then square root of n. We can proof by contradiction that if both numbers are greater then square root of n the their product is not less than n.
So, you can take all integers a up to sqrt(n – 1) instead of all integers up to n. For each a, count the number of b >= a such that a * b < n. Then multiply the result by two to count the pair (b, a) for each pair (a, b) you saw. After that, subtract the integer part of sqrt(n – 1) to ensure the pairs (a, a) were counted exactly once.
Below is the implementation of the above approach:
Time Complexity: O(N*sqrt(N))
GeeksforGeeks has prepared a complete interview preparation course with premium videos, theory, practice problems, TA support and many more features. Please refer Placement 100 for details
- Count number of ordered pairs with Even and Odd Product
- Count number of ordered pairs with Even and Odd Sums
- Count ordered pairs of positive numbers such that their sum is S and XOR is K
- Count all distinct pairs with product equal to K
- Count unordered pairs (i,j) such that product of a[i] and a[j] is power of two
- Count pairs of numbers from 1 to N with Product divisible by their Sum
- Count of pairs upto N such whose LCM is not equal to their product for Q queries
- Count of pairs in an array whose product is a perfect square
- Count of pairs in an array such that the highest power of 2 that divides their product is 1
- Count pairs in Array whose product is a Kth power of any positive integer
- Count of index pairs in array whose range product is a positive integer
- Find the number of ordered pairs such that a * p + b * q = N, where p and q are primes
- Find the Kth pair in ordered list of all possible sorted pairs of the Array
- Number of Co-prime pairs from 1 to N with product equals to N
- Number of pairs in an array having sum equal to product
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to firstname.lastname@example.org. 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.