Difference between revisions of "2021 AMC 10A Problems/Problem 11"

Revision as of 19:45, 5 May 2021

Problem

For which of the following integers $b$ is the base- $b$ number $2021_b - 221_b$ not divisible by $3$ ?

$\textbf{(A)} ~3 \qquad\textbf{(B)} ~4\qquad\textbf{(C)} ~6\qquad\textbf{(D)} ~7\qquad\textbf{(E)} ~8$

Solution 1

We have $\begin{align*} 2021_b - 221_b &= (2021_b - 21_b) - (221_b - 21_b) \\ &= 2000_b - 200_b \\ &= 2b^3 - 2b^2 \\ &= 2b^2(b-1), \end{align*}$ which is divisible by $3$ unless $b\equiv2\pmod{3}.$ The only choice congruent to $2$ modulo $3$ is $\boxed{\textbf{(E)} ~8}.$

~MRENTHUSIASM

Solution 2 (Easy)

Vertically subtracting $2021_b - 221_b$ we see that the ones place becomes 0, and so does the $b^1$ place. Then, we perform a carry (make sure the carry is in base $b$ !). Let $b-2 = A$ . Then, we have our final number as $1A00_b$

Now, when expanding, we see that this number is simply $b^3 - (b - 2)^2$ .

Now, notice that the final number will only be congruent to $b^3-(b-2)^2\equiv0\pmod{3}$ if either $b\equiv0\pmod{3}$ , or if $b\equiv1\pmod{3}$ (because note that $(b - 2)^2$ would become $\equiv1\pmod{3}$ , and $b^3$ would become $\equiv1\pmod{3}$ as well, and therefore the final expression would become $1-1\equiv0\pmod{3}$ . Therefore, $b$ must be $\equiv2\pmod{3}$ . Among the answers, only 8 is $\equiv2\pmod{3}$ , and therefore our answer is $\boxed{\textbf{(E)} ~8}.$

- icecreamrolls8

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 3

https://youtu.be/zYIuBXDhJJA

~savannahsolver

Video Solution by TheBeautyofMath

https://youtu.be/t-EEP2V4nAE

~IceMatrix

@@ Line 17: / Line 17: @@
 ==Solution 2 (Easy)==
-Vertically subtracting <cmath>2021_b - 221_b</cmath> we see that the ones place becomes 0, and so does the <math>b^1</math> place. Then, we perform a carry (make sure the carry is in <math>base b</math>!). Let <math>b-2 = A</math>. Then, we have our final number as <cmath>1A00_b</cmath>
+Vertically subtracting <cmath>2021_b - 221_b</cmath> we see that the ones place becomes 0, 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 8 is <math>\equiv\pmod{3}</math>, and therefore our answer is <math>\boxed{\textbf{(E)} ~8}.</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 8 is <math>\equiv2\pmod{3}</math>, and therefore our answer is <math>\boxed{\textbf{(E)} ~8}.</math>
 - icecreamrolls8

2021 AMC 10A (Problems • Answer Key • Resources)
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