Profile

Cover photo
Joaquim Marques
Works at Agrupamento de Escolas João dä Silva Correia
177 followers|34,778 views
AboutPostsPhotosVideos

Stream

Joaquim Marques

Shared publicly  - 
 
 
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).
3 comments on original post
1
Add a comment...

Joaquim Marques

Shared publicly  - 
 
Cursos on-line. Líderes en formación para empresas mediante metodología e-learning. Planes de formación, instalación de plataforma virtual, creación de cursos a medida
1
Add a comment...
Have him in circles
177 people
Rogério Correia's profile photo
Luis Pais's profile photo
Kamini Nirula's profile photo
Ana Sofia's profile photo
Laura Smith's profile photo
Pedro Magalhães's profile photo
David Neagley's profile photo
Susana Nunes's profile photo
Jason Abercrombie's profile photo

Joaquim Marques

Shared publicly  - 
 
Cursos on-line. Líderes en formación para empresas mediante metodología e-learning. Planes de formación, instalación de plataforma virtual, creación de cursos a medida
1
Add a comment...
People
Have him in circles
177 people
Rogério Correia's profile photo
Luis Pais's profile photo
Kamini Nirula's profile photo
Ana Sofia's profile photo
Laura Smith's profile photo
Pedro Magalhães's profile photo
David Neagley's profile photo
Susana Nunes's profile photo
Jason Abercrombie's profile photo
Basic Information
Gender
Male
Work
Employment
  • Agrupamento de Escolas João dä Silva Correia
    Professor, 2015 - present