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Difference between revisions of "2000 AMC 12 Problems/Problem 4"

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The last digit to appear in the units position of a number in the Fibonacci sequence is <math> 6 \Rightarrow C </math>.
 
The last digit to appear in the units position of a number in the Fibonacci sequence is <math> 6 \Rightarrow C </math>.
  
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== See Also ==  
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== See also ==
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[[2000 AMC 12 Problems]]
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* [[2000 AMC 12 Problems]]
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*[[2000 AMC 12/Problem 3|Previous Problem]]
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*[[2000 AMC 12/Problem 5|Next problem]]
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[[Category:Introductory Combinatorics Problems]]

Revision as of 19:23, 5 November 2006

Problem

The Fibonacci sequence $1,1,2,3,5,8,13,21,\ldots$ starts with two 1s, 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

Looking at the Fibonacci Sequence in $\bmod{10}$:

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

The last digit to appear in the units position of a number in the Fibonacci sequence is $6 \Rightarrow C$.

See also

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