829. Consecutive Numbers Sum

Question

Given a positive integer N, how many ways can we write it as a sum of consecutive positive integers?

Example 1:

Input: 5
Output: 2
Explanation: 5 = 5 = 2 + 3
Example 2:

Input: 9
Output: 3
Explanation: 9 = 9 = 4 + 5 = 2 + 3 + 4
Example 3:

Input: 15
Output: 4
Explanation: 15 = 15 = 8 + 7 = 4 + 5 + 6 = 1 + 2 + 3 + 4 + 5
Note: 1 <= N <= 10 ^ 9.

Solution:

  • Method 1: Math
    • We assume the first term is x and totally m terms.
    • 2xm + m(m - 1) = 2N => we try all possible m and find if there is a valid x. 
      class Solution {
  public int consecutiveNumbersSum(int N) {
      int res = 0;
      for(int m = 1;;m++){
          int mx = 2 * N - m *(m - 1);
          if(mx <= 0) break;
          else if(mx % (2 * m) == 0) res++;
      }
      return res;
  }
}