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Difference between revisions of "2021 Fall AMC 12A Problems/Problem 4"

(Combined solutions. Credits given to both authors.)
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~Steven Chen (www.professorchenedu.com)
 
~Steven Chen (www.professorchenedu.com)
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==Solution 2==
 +
<math>202100 \implies</math> divisible by <math>2</math>.
 +
 +
<math>202101 \implies</math> divisible by <math>3</math>.
 +
 +
<math>202102 \implies</math> divisible by <math>2</math>.
 +
 +
<math>202103 \implies</math> divisible by <math>11</math>.
 +
 +
<math>202104 \implies</math> divisible by <math>2</math>.
 +
 +
<math>202105 \implies</math> divisible by <math>5</math>.
 +
 +
<math>202106 \implies</math> divisible by <math>2</math>.
 +
 +
<math>202107 \implies</math> divisible by <math>3</math>.
 +
 +
<math>202108 \implies</math> divisible by <math>2</math>.
 +
 +
This leaves only <math>A=\boxed{\textbf{(E) }9}</math>.
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 +
~wamofan
  
 
==See Also==
 
==See Also==

Revision as of 22:48, 16 January 2022

The following problem is from both the 2021 Fall AMC 10A #5 and 2021 Fall AMC 12A #4, so both problems redirect to this page.

Problem

The six-digit number $\underline{2}\,\underline{0}\,\underline{2}\,\underline{1}\,\underline{0}\,\underline{A}$ is prime for only one digit $A.$ What is $A?$

$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 9$

Solution

First, modulo $2$ or $5$, $\underline{20210A} \equiv A$. Hence, $A \neq 0, 2, 4, 5, 6, 8$.

Second modulo $3$, $\underline{20210A} \equiv 2 + 0 + 2 + 1 + 0 + A \equiv 5 + A$. Hence, $A \neq 1, 4, 7$.

Third, modulo $11$, $\underline{20210A} \equiv A + 1 + 0 - 0 - 2 - 2 \equiv A - 3$. Hence, $A \neq 3$.

Therefore, the answer is $\boxed{\textbf{(E) }9}$.

~NH14

~Steven Chen (www.professorchenedu.com)

Solution 2

$202100 \implies$ divisible by $2$.

$202101 \implies$ divisible by $3$.

$202102 \implies$ divisible by $2$.

$202103 \implies$ divisible by $11$.

$202104 \implies$ divisible by $2$.

$202105 \implies$ divisible by $5$.

$202106 \implies$ divisible by $2$.

$202107 \implies$ divisible by $3$.

$202108 \implies$ divisible by $2$.

This leaves only $A=\boxed{\textbf{(E) }9}$.

~wamofan

See Also

2021 Fall AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
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 4 		Followed by
Problem 6
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. AMC logo.png

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