2000 AMC 10 Problems/Problem 6

Problem

The Fibonacci sequence $1, 1, 2, 3, 5, 8, 13, 21, \ldots$ starts with two $1$ s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?

$\mathrm{(A)}\ 0 \qquad\mathrm{(B)}\ 4 \qquad\mathrm{(C)}\ 6 \qquad\mathrm{(D)}\ 7\qquad\mathrm{(E)}\ 9$

Solution

The pattern of the units digits are

$1,1,2,3,5,8,3,1,4,5,9,4,3,7,0,7,7,4,1,5,6$

In order of appearance:

$1,2,3,5,8,4,9,7,0,6$ .

$6$ is the last.

$\boxed{\text{C}}$

2000 AMC 10 (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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2000 AMC 10 Problems/Problem 6

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