aptitude, reasoning, Sqlite and problem solving: iven an integer N, the task is to find the sum of Greatest Common Divisors of all numbers up to N with N itself.
Examples:
Input: N = 12
Output: 40
Explanation:
GCD of [1, 12] = 1, [2, 12] = 2, [3, 12] = 3, [4, 12] = 4, [5, 12] = 1, [6, 12] = 6, [7, 12] = 1, [8, 12] = 4, [9, 12] = 3, [10, 12] = 2, [11, 12] = 1, [12, 12] = 12. The sum is (1 + 2 + 3 + 4 + 1 + 6 + 1 + 4 + 3 + 2 + 1 + 12) = 40.
Input: N = 2
Output: 3
Explanation:
GCD of [1, 2] = 1, [2, 2] = 2 and their sum is 3.
Naive Approach: A simple solution is to iterate over all numbers from 1 to N and find their gcd with N itself and keep on adding them.
Time Complexity: O(N * log N)
Efficient Approach:
To optimize the above-mentioned approach, we need to observe that GCD(i, N) gives one of the divisors of N. So, instead of running a loop from 1 to N, we can check for each divisor of N how many numbers are there with GCD(i, N) same as that divisor.
Illustration:
For example N = 12, its divisors are 1, 2, 3, 4, 6, 12.
Numbers in range [1, 12] whose GCD with 12 is:
1 are {1, 5, 7, 11}
2 are {2, 10}
3 are {3, 9}
4 are {4, 8}
6 is {6}
12 is {12}
So answer is; 1*4 + 2*2 + 3*2 + 4*2 + 6*1 + 12*1 = 40.
So we have to find the number of integers from 1 to N with GCD d, where d is a divisor of N. Let us consider x1, x2, x3, …. xn as the different integers from 1 to N such that their GCD with N is d.
Since, GCD(xi, N) = d then GCD(xi/d, N/d) = 1
So, the count of integers from 1 to N whose GCD with N is d is Euler Totient Function of (N/d).
