0 As Divisor
Any Fibonacci number may be divided by 0, but this should not be confused with the multiplicative application of dividing by zero (considered as a value) which produces an infinitely large result and is usually forbidden.
In Fibonacci division, the divisor (Y, B, and B') is considered as a "times" function and only the dividend (X, A, and A') is considered as a value. This distinction will not, in general, produce different results in Fibonacci division, but in division by zero, certains differences will be observed.

EXAMPLE : 4181 divided into 0 parts is :
         -1 part of -2584 plus 1 part of 1597, also
         -1 part of 6765 plus 1 part of 10,946
EXAMPLE : -6765 divided into 0 parts is :
         -1 part of 4181 plus 1 part of -2584
                   as if the dividend had been divided by any other number.

And, as a crowning glory of the Fibonacci system : 0 may even be divided by itself :
EXAMPLE : 0 divided into 0 parts is -1 part of 1 plus 1 part of 1
         for a total of 0 instead of the multiplically correct value of 1 (0 / 0 = 0⁰ = 1)

It is true that division in a closed set of integers (div-mod) is possible with the presence of a remainder. The Fibonacci "division" never has a remainder which makes it quite unique in its own hybrid (addition and multiplication), two-layered (two values of quotients) way.

To See Multiples And Modulos