Difference between revisions of "2000 AMC 12 Problems/Problem 4"
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| + | {{duplicate|[[2000 AMC 12 Problems|2000 AMC 12 #4]] and [[2000 AMC 10 Problems|2000 AMC 10 #6]]}} | ||
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== Problem == | == Problem == | ||
| − | The Fibonacci sequence <math>1,1,2,3,5,8,13,21,\ldots </math> starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten | + | The [[Fibonacci sequence]] <math>1,1,2,3,5,8,13,21,\ldots </math> starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten [[digit]]s is the last to appear in the units position of a number in the Fibonacci sequence? |
| − | <math> \ | + | <math> \textbf{(A)} \ 0 \qquad \textbf{(B)} \ 4 \qquad \textbf{(C)} \ 6 \qquad \textbf{(D)} \ 7 \qquad \textbf{(E)} \ 9 </math> |
== Solution == | == Solution == | ||
| − | + | Note that any digits other than the units digit will not affect the answer. So to make computation quicker, we can just look at the Fibonacci sequence in <math>\bmod{10}</math>: | |
<math>1,1,2,3,5,8,3,1,4,5,9,4,3,7,0,7,7,4,1,5,6,....</math> | <math>1,1,2,3,5,8,3,1,4,5,9,4,3,7,0,7,7,4,1,5,6,....</math> | ||
| − | The last digit to appear in the units position of a number in the Fibonacci sequence is <math> 6 \ | + | The last digit to appear in the units position of a number in the Fibonacci sequence is <math> 6 \Longrightarrow \boxed{\mathrm{C}} </math>. |
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| + | ==Video Solution by Daily Dose of Math== | ||
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| + | https://youtu.be/4dWDOG31tQM?si=Ne3F969T10goxIjC | ||
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| + | ~Thesmartgreekmathdude | ||
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== See also == | == See also == | ||
| − | + | {{AMC12 box|year=2000|num-b=3|num-a=5}} | |
| − | + | {{AMC10 box|year=2000|num-b=5|num-a=7}} | |
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[[Category:Introductory Combinatorics Problems]] | [[Category:Introductory Combinatorics Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 23:39, 14 July 2024
- The following problem is from both the 2000 AMC 12 #4 and 2000 AMC 10 #6, so both problems redirect to this page.
Problem
The Fibonacci sequence 
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?
Solution
Note that any digits other than the units digit will not affect the answer. So to make computation quicker, we can just look at the Fibonacci sequence in 
:
The last digit to appear in the units position of a number in the Fibonacci sequence is 
.
Video Solution by Daily Dose of Math
https://youtu.be/4dWDOG31tQM?si=Ne3F969T10goxIjC
~Thesmartgreekmathdude
See also
| 2000 AMC 12 (Problems • Answer Key • Resources) | |
| Preceded by Problem 3 |
Followed by Problem 5 |
| 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 | |
| All AMC 12 Problems and Solutions | |
| 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 | ||
| All AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
