Question

For each positive integer $k,$ let $S_k$ denote the increasing arithmetic sequence of integers whose first term is $1$ and whose common difference is $k.$ For example, $S_3$ is the sequence $1, 4, 7, \ldots.$ For how many values of $k$ does $S_k$ contain the term $2005?$

Answer

Answer: 12

Concepts:arithmetic sequence factor counting

Difficulty rating: 1840

Solution:

The $n$ th term of $S_k$ is $1 + (n-1)k,$ so $2005$ is a term exactly when $(n-1)k = 2004$ for some positive integer $n,$ that is, exactly when $k$ divides $2004.$ Every divisor works, since $n - 1 = \frac{2004}{k}$ is then a positive integer.

Since $2004 = 2^2 \cdot 3 \cdot 167,$ the number of divisors is $(2+1)(1+1)(1+1) = 12.$

2005 AIME I Problem 2

Answer: 12

Difficulty rating: 1840

Solution:

Problem 2 in Other Years