2022 AMC 10B Problems/Problem 6

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

The $n$ th term of this sequence is $\begin{align*} \left(10^{2n}+10^{2n-1}+\cdots+10^{n}\right)+\left(10^{n}+10^{n-1}+\cdots+10^{0}\right) &= 10^n\left(10^{n}+10^{n-1}+\cdots+10^{0}\right)+\left(10^{n}+10^{n-1}+\cdots+10^{0}\right) \\ &= \left(10^n + 1\right)\left(10^{n}+10^{n-1}+\cdots+10^{0}\right). \end{align*}$ It follows that the first ten terms are divisible by $11,101,1001,\ldots,$ respectively.

Therefore, there are \boxed{\textbf{(A) } 0} prime numbers in this sequence.

~MRENTHUSIASM

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
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2022 AMC 10B Problems/Problem 6

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