Given a number n, find the number of ways to represent this number as a sum of 2 or more consecutive natural numbers.
Input : n = 15 Output : 3 15 can be represented as: 1 + 2 + 3 + 4 + 5 4 + 5 + 6 7 + 8 Input :10 Output :2 10 can only be represented as: 1 + 2 + 3 + 4
We have already discussed one approach in below post.
Count ways to express a number as sum of consecutive numbers
Here a new approach is discussed. Suppose that we are talking about the sum of numbers from X to Y ie [X, X+1, …, Y-1, Y]
Then the arithmetic sum is
If this should be N, then
2N = (Y+X)(Y-X+1)
Note that one of the factors should be even and the other should be odd because Y-X+1 and Y+X should have opposite parity because Y-X and Y+X have the same parity. Since 2N is anyways even, we find the number of odd factors of N.
For example, n = 15 all odd factors of 15 are 1 3 and 5 so the answer is 3.
The Time complexity for this program is O(N^0.5).
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Improved By : manishshaw1