In the ordering $\preceq$ of nonnegative integers by divisibility, 1 is the least element and 0 is the greatest, and we have for instance
$$
1\preceq 2\preceq 6\preceq 12\preceq\dots\preceq 0.$$
In this ordering, gcd is the same as meet (*greatest lower bound*), which is dual to *least upper bound*, which is what boolean OR is for $\{0,1\}$.

So it makes sense if you think of numbers as "degrees of truth", where **multiplicative factors** are evidence of falsehood.

See also: [What is gcd(0,0)?][1]

  [1]: https://math.stackexchange.com/questions/495119/what-is-gcd0-0