Use prefix sums and modular arithmetic: compute cumulative sums modulo k and track the first index where each remainder appears. If the same remainder appears again at least two indices apart, a subarray with sum divisible by k exists.
Pro tip: Handle edge cases like k=0 and negative numbers by normalizing remainders; also mention that the subarray must have length at least 2, so check index difference > 1.
Clarify that we need a continuous subarray of length >= 2 whose sum is a multiple of k. Note that k can be negative or zero, and array elements can be negative.
Compute prefix sums modulo k. If two prefix sums have the same remainder, the subarray between them has sum divisible by k.
Use a hash map to store the earliest index for each remainder. Initialize with remainder 0 at index -1 to handle subarrays starting at index 0.
When a remainder repeats, check if the current index minus the stored index is at least 2. If so, return true.
If k=0, check for any zero-sum subarray of length >=2. Normalize negative remainders by adding k. Return false if no valid subarray found.
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