2022 AMC 12B Problems/Problem 3

Revision as of 17:18, 17 November 2022 by Bxiao31415 (talk | contribs) (Solution 1)

Problem

How many of the first ten numbers of the sequence $121$, $11211$, $1112111$, ... are prime numbers? $\text{(A) } 0 \qquad \text{(B) }1 \qquad \text{(C) }2 \qquad \text{(D) }3 \qquad \text{(E) }4$

Solution 1

Let $P(a,b)$ denote the digit $a$ written $b$ times and let $\overline{a_1a_2\cdots a_n}$ denote the concatenation of $a_1$, $a_2$, ..., $a_n$. Observe that \[\overline{P(1,n) 2 P(1,n)} = \overline{P(1,n+1)P(0,n)} + P(1,n+1) = P(1,n+1) \cdot 10^n + P(1,n+1) = (P(1,n+1))(10^n + 1).\] Clearly, both terms are larger than $1$ since $n \geq 1$, hence all the numbers of the sequence are $\fbox{0(A)}$, and we're done!

~Bxiao31415

See Also

2022 AMC 12B (ProblemsAnswer KeyResources)
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

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