2022 AMC 10B Problem 6

Question

How many of the first ten numbers of the sequence

121, 11211, 1112111, \ldots

are prime numbers?

0

1

2

3

4

Express the

n

th term as the sum of two overlapping blocks of

n+1

ones

Each term in the sequence has an obvious factorization pattern

We claim that none of these numbers can ever be prime.

We prove this claim by noticing that the

n

th number is

\sum_{k=0}^{2n} 10^k + 10^n = \sum_{k=0}^{n} 10^k +

\sum_{k=n}^{2n} 10^k =\sum_{k=0}^{n} 10^k + \sum_{k=0}^{n} 10^k \cdot 10^n

= (10^n+1)(\sum_{k=0}^{n} 10^k).

This shows that the number can be written as the product of two numbers greater than

1,

so there are no primes.

Thus, the answer is A .

Problem 6 in other years