The Modonacci Puzzle


The Fibonacci sequence begins with 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... with each new term formed by adding the two preceding ones.

The "Modonacci" number M is formed by starting with a decimal point, and then appending the last digit of each entry in the Fibonacci sequence.

Thus, the first ten decimal places are:

M = 0.1123583145...


Puzzle 1: Prove that M is a rational number.


Puzzle 2: Since M is a rational number, it can be written as a ratio P/Q, where P and Q are integers with no common factor. What are the values of P and Q?


I give up, show me the solution.


This puzzle was kindly supplied by Scott Hansen.


Last revised on 07 March 2011.