Smallest Integer Divisible by K - Solution & Explanation

This approach leverages the properties of remainders in modular arithmetic. We try to construct a number which consists of only '1's and is divisible by k. We start with the number 1 and find its remainder with k. Then, iteratively, we add another 1 at the right (equivalent to multiply by 10 and add 1) and find the new remainder. We repeat this process until the remainder becomes 0 or a repetition is detected.

If we encounter a remainder that we have seen before, it indicates a cycle, and there is no such number n that will satisfy the condition, hence return -1.

We start with a remainder of 0 and iterate up to k times, constructing the number step-by-step by updating the remainder. The formula remainder = (remainder * 10 + 1) % k helps us construct the number in the form of '111...1' without actually building this number explicitly. If we find a remainder of 0, the length of such a number is returned. If a cycle is detected, meaning all possible remainders are tried without success, -1 is returned.

This approach still utilizes the modulus technique but introduces a digit construction method. Essentially, instead of just checking the length, we can construct the number step-by-step while simultaneously detecting cycles. The cycle detection ensures we do not repeat remainder calculations unnecessarily, identifying when no solution exists faster.

The goal remains to build a number consisting only of '1's that is divisible by k. The difference here is the emphasis on cycle and digit tracking during construction.

Here, we include an additional array to track each remainder used. A cycle is detected if the same remainder appears twice before finding a solution. Using visited_remainders, we store each state and verify conditions for possible cycles. This strategy optimizes cycle detection and avoids unnecessary computations.