Check if chords of a Circle are symmetric after some rotation

Given two Integers N and M, N indicating equidistant points on circumference of a circle and M indicating number of chords formed with those points. Also given is a vector of pairs C containing position of chords. The task is to rotate the circle by any degree, say X, where 0 < X < 360, and check if the chords of are still symmetric to the original circle.

Example: 

Input: N = 12, M = 6, C = {{1, 3}, {3, 7}, {5, 7}, {7, 11}, {9, 11}, {11, 3}};
Output: YES

👁 Image

👁 Image

Input: N = 10, M = 3,  C = {{1, 2}, {3, 2}, {7, 2}}
Output: NO

👁 Image

Naive Approach: Rotate for every distance K in the range[1, N] and check for each point [a, b] if the rotated point [a + K, b + K] exits. 
If there exists any k then print YES else print NO
Time Complexity: O(N*M)

Efficient Approach: It is enough to check for the divisors of N. 
Let us suppose if we rotate the image by K units then the whole image will be divided into N/K blocks. Then if K is not a divisor of N, there will be an asymmetric block of length less than K and the image will never be symmetric to the original figure.
So calculate all the divisors of N and check for each chord the rotated chord exists or not.

Below is the implementation of the above approach:

YES

Time Complexity: O(M*sqrt(N)*log M)
Space Complexity: O(M)