The bitwise AND of a set of numbers will have a bit set to 1 in a particular position if and only if all numbers in the set have that bit set to 1. Thus, the problem reduces to finding the maximum number of elements that share at least one common bit position.

For each bit position (0 to 30, since integers are 32-bit), count how many numbers in the array have that bit set. The maximum count across all bit positions is the answer, as it represents the largest subset where all elements share at least one common bit.

To optimize, iterate through each bit position, and for each number in the array, check if the bit is set using `(num >> bit) & 1`. Maintain a count for each bit and return the maximum count found. This approach runs in O(32 * n) time, which is efficient for large arrays.