Difference between revisions of "2021 AMC 10A Problems/Problem 11"
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<math>\textbf{(A)} ~3 \qquad\textbf{(B)} ~4\qquad\textbf{(C)} ~6\qquad\textbf{(D)} ~7\qquad\textbf{(E)} ~8</math> | <math>\textbf{(A)} ~3 \qquad\textbf{(B)} ~4\qquad\textbf{(C)} ~6\qquad\textbf{(D)} ~7\qquad\textbf{(E)} ~8</math> | ||
− | ==Solution== | + | ==Solution 1 (Factor)== |
− | We have <cmath>2021_b - 221_b = 2000_b - 200_b = 2b^3 - 2b^2 = 2b^2(b-1) | + | We have |
+ | <cmath>\begin{align*} | ||
+ | 2021_b - 221_b &= (2021_b - 21_b) - (221_b - 21_b) \\ | ||
+ | &= 2000_b - 200_b \\ | ||
+ | &= 2b^3 - 2b^2 \\ | ||
+ | &= 2b^2(b-1), | ||
+ | \end{align*}</cmath> | ||
+ | which is divisible by <math>3</math> <b><i>unless</i></b> <math>b\equiv2\pmod{3}.</math> The only choice congruent to <math>2</math> modulo <math>3</math> is <math>\boxed{\textbf{(E)} ~8}.</math> | ||
~MRENTHUSIASM | ~MRENTHUSIASM | ||
+ | |||
+ | ==Solution 2 (Vertical Subtraction)== | ||
+ | Vertically subtracting <math>2021_b - 221_b,</math> we see that the ones place becomes <math>0,</math> and so does the <math>b^1</math> place. Then, we perform a carry (make sure the carry is in base <math>b</math>). Let <math>b-2 = A.</math> Then, we have our final number as <cmath>1A00_b.</cmath> | ||
+ | Now, when expanding, we see that this number is simply <math>b^3 - (b - 2)^2.</math> | ||
+ | |||
+ | Now, notice that the final number will only be congruent to <cmath>b^3-(b-2)^2\equiv0\pmod{3}.</cmath> If either <math>b\equiv0\pmod{3},</math> or if <math>b\equiv1\pmod{3}</math> (because note that <math>(b - 2)^2</math> would become <math>\equiv1\pmod{3},</math> and <math>b^3</math> would become <math>\equiv1\pmod{3}</math> as well, and therefore the final expression would become <math>1-1\equiv0\pmod{3}.</math> Therefore, <math>b</math> must be <math>\equiv2\pmod{3}.</math> Among the answers, only <math>8</math> is <math>\equiv2\pmod{3},</math> and therefore our answer is <math>\boxed{\textbf{(E)} ~8}.</math> | ||
+ | |||
+ | ~icecreamrolls8 | ||
+ | |||
+ | ==Solution 3 (Answer Choices)== | ||
+ | By the definition of bases, we have <cmath>2021_b - 221_b = \left(2b^3+2b+1\right) - \left(2b^2+2b+1\right).</cmath> | ||
+ | For values <math>b_1</math> and <math>b_2</math> such that <math>b_1\equiv b_2\pmod{3},</math> we get <cmath>\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}.</cmath> | ||
+ | Note that answer choices <math>\textbf{(A)},\textbf{(B)},\textbf{(C)},\textbf{(D)},\textbf{(E)}</math> are congruent to <math>0,1,0,1,2</math> modulo <math>3,</math> respectively. So, <math>\textbf{(A)}</math> and <math>\textbf{(C)}</math> are either both correct or both incorrect. Since there is only one correct answer, <math>\textbf{(A)}</math> and <math>\textbf{(C)}</math> are both incorrect. Similarly, <math>\textbf{(B)}</math> and <math>\textbf{(D)}</math> are both incorrect. This leaves us with <math>\boxed{\textbf{(E)} ~8},</math> the answer choice with a unique residue modulo <math>3.</math> | ||
+ | |||
+ | ~[[User:emerald_block|emerald_block]] ~MRENTHUSIASM | ||
+ | |||
+ | ==Video Solution by Omega Learn== | ||
+ | https://youtu.be/CPYCUnL_Yc0 | ||
+ | |||
+ | ~ pi_is_3.14 | ||
+ | |||
+ | ==Video Solution (Simple and Quick)== | ||
+ | https://youtu.be/1TZ1uI9z8fU | ||
+ | |||
+ | ~ 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== | ||
+ | https://youtu.be/zYIuBXDhJJA | ||
+ | |||
+ | ~savannahsolver | ||
+ | |||
+ | ==Video Solution by TheBeautyofMath== | ||
+ | https://youtu.be/t-EEP2V4nAE | ||
+ | |||
+ | ~IceMatrix | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC10 box|year=2021|ab=A|num-b=10|num-a=12}} | ||
+ | {{MAA Notice}} |
Latest revision as of 11:05, 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
Now, when expanding, we see that this number is simply
Now, notice that the final number will only be congruent to 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
For values
and
such that
we get
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.