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Joaquim Marques
Works at Agrupamento de Escolas João da Silva Correia
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Joaquim Marques

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Theorem (I learned recently, via +Matt Enlow then Mike Lawler): if a Fibonacci number is a multiple of 9, it's a multiple of 8.

Proof: consider them mod 72. They go
0,1,1,2,3,5,8,13,21,34,55,17,0,17,...
and on the second go-round after 0 the sequence is 17x the first go-round. That 17 doesn't affect divisibility by 8 or 9.
The only multiple of 9 in that list is 0. So divisible by 9 => divisible by 72 => divisible by 8.
(In fact 17^2 = 1 mod 72 so the third go-round is where we get back to 0,1,... but this isn't important for the argument.)

EDIT: this calculation (including the comment at the end) also shows the converse is almost true; if a Fibonacci number is a multiple of 8, then it's either a multiple of 9 or off by one (because of the 0, 8, and 17*8 = 64).
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  • Agrupamento de Escolas João da Silva Correia
    Professor, 2015 - present