2022 AMC 12B Problems/Problem 3

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!

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2022 AMC 12B Problems/Problem 3

Problem

Solution 1