2005 AIME I Problems/Problem 2
Problem
For each positive integer , let denote the increasing arithmetic sequence of integers whose first term is 1 and whose common difference is . For example, is the sequence For how many values of does contain the term 2005?
Solution
Suppose that the th term of the sequence is 2005. Then so . The ordered pairs of positive integers that satisfy the last equation are ,, , , , , ,, , , and , and each of these gives a possible value of . Thus the requested number of values is 12.
Alternatively, notice that the formula for the number of divisors states that there are divisors of .
See also
2005 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |