When Digits Change: Binary Representation and the Arithmetic Beneath It

Binary notation makes digit changes exact arithmetic operations: removing a 1 subtracts a power of two. This essay examines how such changes preserve, destroy, or establish divisibility. Repeating positional weights explain single-deletion rules, minimum deletion counts, and the different requirements imposed by factors of the same number. Exact formulas distinguish arithmetic constraints from the number of available positions, and mixed binary strings extend the analysis to additions and movements of digits. Continuing the representational inquiry of WidiVision, the essay offers an accessible synthesis with a technical core, grounded in established work on digitally delicate primes and sturdy numbers. It makes no claim to a new general factoring or primality algorithm.