Problem #5

The number 9 can be written as the sum of two consecutive positive integers: 9 = 4 + 5. It can be written as the sum of at least two consecutive positive integers in exactly two ways: 9 = 4 + 5 and 9 = 2 + 3 + 4. Is there a positive integer that can be written as the sum of 2003 consecutive positive integers and can be written as the sum of at least two consecutive positive integers in exactly 2003 ways?

What if 2003 is replaced by 2002?

Source: A problem proposed, but not used, for the International Mathematical Olympiad