How does "magic number" algorithm work

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Would anyone happen to have a pointer to a proof of how the "magic
number" algorithm for finding a subnet  number's "interesting" octet
decimal value works?  I.e., WHY does it work, mathematically speaking?

Algorithm is:  subtract the mask's "interesting" (non-255) decimal
number from 256, which yields the "magic number".  Then find the
largest multiple of the "magic number" that is less than the
"interesting" decimal value in question--this is the "interesting"
decimal number for the subnet address.

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