Difference between revisions of "2021 AMC 10A Problems/Problem 11"
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| + | ==Video Solution by Omega Learn== | ||
| + | https://youtu.be/CPYCUnL_Yc0 | ||
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| + | ~ pi_is_3.14 | ||
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https://youtu.be/t-EEP2V4nAE | https://youtu.be/t-EEP2V4nAE | ||
~IceMatrix | ~IceMatrix | ||
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==See Also== | ==See Also== | ||
{{AMC10 box|year=2021|ab=A|num-b=10|num-a=12}} | {{AMC10 box|year=2021|ab=A|num-b=10|num-a=12}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 10:05, 16 June 2023
Contents
Problem
For which of the following integers 
is the base- number 
not divisible by 
?
Solution 1 (Factor)
We have
which is divisible by unless 
The only choice congruent to 
modulo is 
~MRENTHUSIASM
Solution 2 (Vertical Subtraction)
Vertically subtracting 
we see that the ones place becomes 
and so does the 
place. Then, we perform a carry (make sure the carry is in base ). Let 
Then, we have our final number as ![\[1A00_b.\]](http://latex.artofproblemsolving.com/5/7/b/57bf219eb587fba6c8f9ace0e2bafa34db830d5c.png)
Now, when expanding, we see that this number is simply 
Now, notice that the final number will only be congruent to ![\[b^3-(b-2)^2\equiv0\pmod{3}.\]](http://latex.artofproblemsolving.com/c/4/7/c47bd25aaeb0065f35b3577c1189e858ee01ad39.png)
If either 
or if 
(because note that 
would become 
and 
would become 
as well, and therefore the final expression would become 
Therefore, must be 
Among the answers, only 
is 
and therefore our answer is
~icecreamrolls8
Solution 3 (Answer Choices)
By the definition of bases, we have ![\[2021_b - 221_b = \left(2b^3+2b+1\right) - \left(2b^2+2b+1\right).\]](http://latex.artofproblemsolving.com/a/a/d/aad0bb9e5ffe7879965f9063459cbb542086cc8b.png)
For values 
and 
such that 
we get ![\[\left(2b_1^3+2b_1+1\right) - \left(2b_1^2+2b_1+1\right) \equiv \left(2b_2^3+2b_2+1\right) - \left(2b_2^2+2b_2+1\right) \pmod{3}.\]](http://latex.artofproblemsolving.com/3/3/5/335e629d4560a0c08ee369a79cab848a20ee3c63.png)
Note that answer choices 
are congruent to 
modulo 
respectively. So, 
and 
are either both correct or both incorrect. Since there is only one correct answer, and are both incorrect. Similarly, 
and 
are both incorrect. This leaves us with 
the answer choice with a unique residue modulo 
~emerald_block ~MRENTHUSIASM
Video Solution by Omega Learn
~ pi_is_3.14
Video Solution (Simple and Quick)
~ Education, the Study of Everything
Video Solution
https://www.youtube.com/watch?v=XBfRVYx64dA&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=10
~North America Math Contest Go Go Go
Video Solution
~savannahsolver
Video Solution by TheBeautyofMath
~IceMatrix
See Also
| 2021 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 10 |
Followed by Problem 12 | |
| 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. 
