Consider circular track with n points marked as 1, 2, …n. A person is initially placed on point k. The person moves m > 0, slot forward (in circular way) in each step. Find the minimum number of steps required so that the person reaches initial point k.
Input : n = 9, k = 2, m = 6 Output : 3 Explanation : Sequence of moves is 2 => 8 => 5 => 2 Input : n = 6, k = 3, m = 2 Output : 3
Naive Approach : Initialize a counter ‘i’ with ‘k’ and ‘count’ = 0. Further for each iteration increment ‘count’ add ‘m’ to ‘i’. Take its modulus with n i.e. i=((i+m)%n), if i > n. If i becomes equal to k then count will be our answer.
Time complexity: O(n).
Efficient Approach: We find GCD(n, m) and then divide n by GCD(n, m). That will be our answer. This can be explained as:
Think of n and m as per question now as we know that gcd(n, m) must divide n and the quotient tells us that after how many successive jumps(addition) of m numbers from starting position(say 0) we again reach the starting position.
Note: In circular arrangement of n numbers nth and 0th position are same.
Time Complexity: O(log(n))
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