Difference between revisions of "2006 Alabama ARML TST Problems/Problem 8"
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==Problem== | ==Problem== | ||
| − | A bored mathematician has his computer calculate 1000 consecutive terms in the Fibonacci sequence. He notes that the smallest of the numbers is a multiple of 7. How many of the other 999 Fibonacci numbers are multiples of 7? | + | A bored [[mathematician]] has his computer calculate 1000 consecutive terms in the [[Fibonacci sequence]]. He notes that the smallest of the numbers is a [[multiple]] of 7. How many of the other 999 Fibonacci numbers are multiples of 7? |
==Solution== | ==Solution== | ||
Latest revision as of 15:43, 17 April 2008
Problem
A bored mathematician has his computer calculate 1000 consecutive terms in the Fibonacci sequence. He notes that the smallest of the numbers is a multiple of 7. How many of the other 999 Fibonacci numbers are multiples of 7?
Solution
We list the Fibonacci numbers mod 7:
1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1, 0, 1, 1, ....
So the Fibonacci sequence repeats every 16, and for every 8 numbers, there is one 0. 
, but we don't count the first one. 
See also
| 2006 Alabama ARML TST (Problems) | ||
| Preceded by: Problem 7 |
Followed by: Problem 9 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
