Difference between revisions of "2021 Fall AMC 12A Problems/Problem 10"
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~Wilhelm Z | ~Wilhelm Z | ||
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==Video Solution (HOW TO THINK CREATIVELY!!!)== | ==Video Solution (HOW TO THINK CREATIVELY!!!)== | ||
https://youtu.be/Mv38a8oanFk | https://youtu.be/Mv38a8oanFk |
Revision as of 21:24, 15 June 2023
- The following problem is from both the 2021 Fall AMC 10A #12 and 2021 Fall AMC 12A #10, so both problems redirect to this page.
Contents
Problem
The base-nine representation of the number is
What is the remainder when
is divided by
Solution 1 (Modular Arithmetic)
Recall that We expand
by the definition of bases:
~Aidensharp ~Kante314 ~MRENTHUSIASM
Solution 2 (Powers of 9)
We need to first convert into a regular base-
number:
Now, consider how the last digit of changes with changes of the power of
Note that if
is odd, then
On the other hand, if
is even, then
Therefore, we have
Note that for the odd case,
may simplify the process further, as given by Solution 1.
~Wilhelm Z
Video Solution (HOW TO THINK CREATIVELY!!!)
https://youtu.be/Mv38a8oanFk
~Education, the Study of Everything
Video Solution
https://youtu.be/SCGzEOOICr4?t=101
Video Solution
https://youtu.be/zq3UPu4nwsE?t=358
https://youtu.be/wlDlByKI7A8?t=885
Video Solution by WhyMath
~savannahsolver
See Also
2021 Fall AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 9 |
Followed by Problem 11 |
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 |
2021 Fall AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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.