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2022 AMC 10B Problems/Problem 6: Difference between revisions

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{{duplicate|[[2022 AMC 10B Problems/Problem 6|2022 AMC 10B #6]] and [[2022 AMC 12B Problems/Problem 3|2022 AMC 12B #3]]}}
==Problem==
==Problem==
How many of the first ten numbers of the sequence <math>121, 11211, 1112111, \ldots</math> are prime numbers?
How many of the first ten numbers of the sequence <math>121, 11211, 1112111, \ldots</math> are prime numbers?
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<math>\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4</math>
<math>\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4</math>


==Solution==
==Solution 1 (Generalization)==
The <math>n</math>th term of this sequence is
The <math>n</math>th term of this sequence is
<cmath>\begin{align*}
<cmath>\sum_{k=n}^{2n}10^k + \sum_{k=0}^{n}10^k = 10^n\sum_{k=0}^{n}10^k + \sum_{k=0}^{n}10^k = \left(10^n+1\right)\sum_{k=0}^{n}10^k.</cmath>
\left(10^{2n}+10^{2n-1}+\cdots+10^{n}\right)+\left(10^{n}+10^{n-1}+\cdots+10^{0}\right) &= 10^n\left(10^{n}+10^{n-1}+\cdots+10^{0}\right)+\left(10^{n}+10^{n-1}+\cdots+10^{0}\right) \\
&= \left(10^n + 1\right)\left(10^{n}+10^{n-1}+\cdots+10^{0}\right).
\end{align*}</cmath>
It follows that the terms are
It follows that the terms are
<cmath>\begin{align*}
<cmath>\begin{align*}
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~MRENTHUSIASM
~MRENTHUSIASM


==Solution 2 (If you don't see the above/are running low on time)==
==Solution 2 (Detailed Explanation of Solution 1)==
 
Denote this sequence as <math>a_{n}</math>, then we can find that
<cmath>\begin{align*}
a_{1} &= 121 = 10^2 + 2\cdot10 + 1 = (10^2 + 10) + (10 + 1), \\
a_{2} &= 11211 = (10^4 + 10^3 + 10^2) + (10^2 + 10 + 1), \\
a_{3} &= 1112111 = (10^6 + 10^5 + 10^4 + 10^3) + (10^3 + 10^2 + 10 + 1), \\
& \ \vdots
\end{align*}</cmath>
So, we can induct that the general term is
<cmath>\begin{align*}
a_n &= (10^{2n} + 10^{2n-1} + \ldots + 10^{n+1} + 10^n) + (10^n + 10^{n-1} + \ldots +10 + 1) \\
&= 10^n\cdot(10^n + 10^{n-1} + \ldots +10 + 1) + (10^n + 10^{n-1} + \ldots +10 + 1) \\
&= \left(10^n+1\right)\sum_{k=0}^{n}10^k.
\end{align*}</cmath>
Therefore, there are <math>\boxed{\textbf{(A) } 0}</math> prime numbers in this sequence.
 
~PythZhou
 
==Solution 3 (Simple Sums)==
 
Observe how
<cmath>\begin{align*}
121 &= 110 + 11, \\
11211 &= 11100 + 111, \\
1112111 &= 1111000 + 1111, \\
& \ \vdots
\end{align*}</cmath>
all take the form of <cmath>\underbrace{111\ldots}_{n+1}\underbrace{00\ldots}_{n} + \underbrace{111\ldots}_{n+1} = \underbrace{111\ldots}_{n+1}(10^{n} + 1).</cmath>
Factoring each of the sums, we have <cmath>11(10+1), 111(100+1), 1111(1000+1), \ldots</cmath> respectively. With each number factored, there are <math>\boxed{\textbf{(A) } 0}</math> primes in the set.
 
~ab2024


Note that it's obvious that 121 is divisible by 11 and 11211 is divisible by 3; therefore, since this an AMC 10 problem 6, we may safely assume that we do not need to check two digit prime divisibility or use obscure theorems; therefore, the answer is <math>\boxed{\textbf{(A)} 0}.</math>
==Solution 4 (Educated Guess)==
 
Note that <math>121</math> is divisible by <math>11</math> and <math>11211</math> is divisible by <math>3</math>. Because this is Problem 6 of the AMC 10, we assume we do not need to check two-digit prime divisibility or use obscure theorems. Therefore, the answer is <math>\boxed{\textbf{(A) } 0}.</math>


~Dhillonr25
~Dhillonr25
==Video Solution(1-16)==
https://youtu.be/SCwQ9jUfr0g
~~Hayabusa1
==Video Solution by Interstigation==
https://youtu.be/_KNR0JV5rdI?t=562
==Video Solution by Math4All999==
https://youtu.be/5QYh3hNaDa0?feature=shared


== See Also ==
== See Also ==
{{AMC10 box|year=2022|ab=B|num-b=5|num-a=7}}
{{AMC10 box|year=2022|ab=B|num-b=5|num-a=7}}
{{AMC12 box|year=2022|ab=B|num-b=2|num-a=4}}
{{MAA Notice}}
{{MAA Notice}}

Latest revision as of 11:18, 20 March 2024

The following problem is from both the 2022 AMC 10B #6 and 2022 AMC 12B #3, so both problems redirect to this page.

Problem

How many of the first ten numbers of the sequence $121, 11211, 1112111, \ldots$ are prime numbers?

$\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4$

Solution 1 (Generalization)

The $n$th term of this sequence is \[\sum_{k=n}^{2n}10^k + \sum_{k=0}^{n}10^k = 10^n\sum_{k=0}^{n}10^k + \sum_{k=0}^{n}10^k = \left(10^n+1\right)\sum_{k=0}^{n}10^k.\] It follows that the terms are \begin{align*} 121 &= 11\cdot11, \\ 11211 &= 101\cdot111, \\ 1112111 &= 1001\cdot1111, \\ & \ \vdots \end{align*} Therefore, there are $\boxed{\textbf{(A) } 0}$ prime numbers in this sequence.

~MRENTHUSIASM

Solution 2 (Detailed Explanation of Solution 1)

Denote this sequence as $a_{n}$, then we can find that \begin{align*} a_{1} &= 121 = 10^2 + 2\cdot10 + 1 = (10^2 + 10) + (10 + 1), \\ a_{2} &= 11211 = (10^4 + 10^3 + 10^2) + (10^2 + 10 + 1), \\ a_{3} &= 1112111 = (10^6 + 10^5 + 10^4 + 10^3) + (10^3 + 10^2 + 10 + 1), \\ & \ \vdots \end{align*} So, we can induct that the general term is \begin{align*} a_n &= (10^{2n} + 10^{2n-1} + \ldots + 10^{n+1} + 10^n) + (10^n + 10^{n-1} + \ldots +10 + 1) \\ &= 10^n\cdot(10^n + 10^{n-1} + \ldots +10 + 1) + (10^n + 10^{n-1} + \ldots +10 + 1) \\ &= \left(10^n+1\right)\sum_{k=0}^{n}10^k. \end{align*} Therefore, there are $\boxed{\textbf{(A) } 0}$ prime numbers in this sequence.

~PythZhou

Solution 3 (Simple Sums)

Observe how \begin{align*} 121 &= 110 + 11, \\ 11211 &= 11100 + 111, \\ 1112111 &= 1111000 + 1111, \\ & \ \vdots \end{align*} all take the form of \[\underbrace{111\ldots}_{n+1}\underbrace{00\ldots}_{n} + \underbrace{111\ldots}_{n+1} = \underbrace{111\ldots}_{n+1}(10^{n} + 1).\] Factoring each of the sums, we have \[11(10+1), 111(100+1), 1111(1000+1), \ldots\] respectively. With each number factored, there are $\boxed{\textbf{(A) } 0}$ primes in the set.

~ab2024

Solution 4 (Educated Guess)

Note that $121$ is divisible by $11$ and $11211$ is divisible by $3$. Because this is Problem 6 of the AMC 10, we assume we do not need to check two-digit prime divisibility or use obscure theorems. Therefore, the answer is $\boxed{\textbf{(A) } 0}.$

~Dhillonr25

Video Solution(1-16)

https://youtu.be/SCwQ9jUfr0g

~~Hayabusa1

Video Solution by Interstigation

https://youtu.be/_KNR0JV5rdI?t=562

Video Solution by Math4All999

https://youtu.be/5QYh3hNaDa0?feature=shared

See Also

2022 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 5 		Followed by
Problem 7
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
2022 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing

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