MathNovatory/Pure Maths/Fibonacci Numbers/Multiples And Modulos


Structural Repeatability

Multiples
All Fibonacci numbers find multiples at a regular interval :
     2 finds a multiple at every 3 numbers (2 3 5 8 13 21 34 55 89 144 ...)
     3 finds a multiple at every 4 numbers (3 5 8 13 21 34 55 89 144 ...)
     5 finds a multiple at every 5 numbers (5 8 13 21 34 55 89 144 233 377 610 ...)
     8 finds a multiple at every 6 numbers (8 13 21 34 55 89 144 ...)
     13 finds a multiple at every 7 numbers (13 21 34 55 89 144 233 377 ...) etc ...

Modulos
Between a Fibonacci number and its first multiple,
     it is interesting to establish the Modulo (M) of each intervening Fibonacci number
          to the base (B) of the original number.
Note all the Fibonacci series in the columns.

B2 3M1 5M1 8M0
B3 5M2 8M2 13M1 21M0
B5 8M3 13M3 21M1 34M-1 55M0
B8 13M5 21M5 34M2 55M-1 89M1 144M0
B13 21M8 34M8 55M3 89M-2 144M1 233M-1 377M0
B21 34M13 55M13 89M5 144M-3 233M2 377M-1 610M1 987M0
B34 55M21 89M21 144M8 233M-5 377M3 610M-2 987M1 1597M-1 2584M0
B55 89M34 144M34 233M13 377M-8 610M5 987M-3 1597M2 2584M-1 4191M1 6785M0
B89 144M55 233M55 377M21 610M-13 987M8 1597M-5 2584M3 4191M-2 6785M1 10946M-1 17711M0