Difference between revisions of "2021 Fall AMC 12A Problems/Problem 10"
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==Solution 2 (Powers of 9)== | ==Solution 2 (Powers of 9)== | ||
− | We need to first convert <math>N</math> into a regular base-<math>10</math> | + | We need to first convert <math>N</math> into a regular base-<math>10</math> number: <cmath>N = 27{,}006{,}000{,}052_9 = 2\cdot9^{10} + 7\cdot9^9 + 6\cdot9^6 + 5\cdot9 + 2.</cmath> |
Now, consider how the last digit of <math>9</math> changes with changes of the power of <math>9:</math> | Now, consider how the last digit of <math>9</math> changes with changes of the power of <math>9:</math> |
Revision as of 20:09, 26 November 2021
- 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.
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
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.