Difference between revisions of "2021 AMC 10A Problems/Problem 11"
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==Solution 3 (Residues)== | ==Solution 3 (Residues)== | ||
− | By definition of bases, | + | 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 two values of <math>b</math> with the same residue modulo <math>3,</math> each of the expressions <math>2b^3+2b+1</math> and <math>2b^2+2b+1</math> have the same residue modulo <math>3.</math> If <math>2021_b - 221_b</math> is divisible by <math>3,</math> then <math>2b^3+2b+1</math> and <math>2b^2+2b+1</math> are both or neither divisible by <math>3.</math> | |
+ | |||
+ | Note that 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. This means <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, we conclude that <math>\textbf{(B)}</math> and <math>\textbf{(D)}</math> are both incorrect. This leaves <math>\boxed{\textbf{(E)} ~8},</math> the answer choice with a unique residue. | ||
~MRENTHUSIASM (revised by [[User:emerald_block|emerald_block]]) | ~MRENTHUSIASM (revised by [[User:emerald_block|emerald_block]]) |
Revision as of 18:50, 22 November 2021
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 (Residues)
By the definition of bases, we have
For two values of with the same residue modulo
each of the expressions
and
have the same residue modulo
If
is divisible by
then
and
are both or neither divisible by
Note that choices are congruent to
modulo
respectively. This means
and
are either both correct or both incorrect. Since there is only one correct answer,
and
are both incorrect. Similarly, we conclude that
and
are both incorrect. This leaves
the answer choice with a unique residue.
~MRENTHUSIASM (revised by emerald_block)
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 3
~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.