Скачать или смотреть Calculate the Sum of All Integers in a Range Divisible by k with O(1) Time Complexity

Learn how to efficiently calculate the sum of all integers in a given range that are divisible by a specified number `k` using a simple formula, all while achieving an astonishing time complexity of O(1).
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How to Calculate the Sum of All Integers in a Range Divisible by k

Are you facing a challenge in calculating the sum of all integers within a certain range that are divisible by a specific integer k? If so, you are not alone. This is a common task in programming, especially when dealing with numeric data. Surprisingly, there's a way to compute this total using a formula that achieves a time complexity of O(1)! Let’s dive into it.

Understanding the Problem

We need to compute the sum of all integers between two endpoints, a and b, that are divisible by a given integer k. A naive approach might involve looping through each integer in the range and checking if it is divisible by k, which would lead to a time complexity of O(n). However, we want our solution to be more efficient.

Optimal Solution

The Formula Approach

Our goal is to find a formula that allows us to perform this calculation quickly without iterating through each number. The solution involves a couple of key steps:

Finding the Start Point: We need to determine the first integer in the range [a, b] that is divisible by k. Depending on a, this might either be a itself or the next larger integer that meets the divisibility condition.

Finding the End Point: Similarly, the last integer in the range that is divisible by k can be found using a simple adjustment to b.

Counting the Values: Once we have both the start and the end points, we calculate how many integers fall within the range that are divisible by k.

Calculating the Sum: Finally, we can apply the arithmetic sum formula to find the total.

Step-by-Step Breakdown

Here’s how this can be implemented in Java:

[[See Video to Reveal this Text or Code Snippet]]

Explanation of Key Parts

Finding start: The calculation a + k - a % k adjusts a to the next multiple of k, if it isn't already one.

Finding end: The formula b - b % k rounds down b to the nearest lower multiple of k.

Counting Numbers: We use integer division to determine how many multiples of k fit into the range from start to end.

Sum Calculation: The arithmetic series formula (start + end) * num / 2 provides the required sum efficiently.

Conclusion

In summary, calculating the sum of integers within a range that are divisible by a specific integer k can efficiently be done using a set formula without extensive loops, achieving O(1) time complexity. By understanding the steps to find the appropriate start and end points in the range, you can apply this method effortlessly in your Java programs. Happy coding!