Find the smallest $n$ such that every subset of $\{1, 2, 3, . . . , 2004 \}$ with $n$ elements contains at least two elements that are relatively prime.
1. We need to find the smallest \( n \) such that every subset of \(\{1, 2, 3, \ldots, 2004\}\) with \( n \) elements contains at least two elements that are relatively prime.
2. Two numbers are relatively prime if their greatest common divisor (gcd) is 1.
3. Consider the set \(\{2, 4, 6, \ldots, 2004\}\). This set contains all even numbers from 1 to 2004, and it has 1002 elements.
4. Any subset of \(\{2, 4, 6, \ldots, 2004\}\) will not contain any two elements that are relatively prime because all elements are even and share a common factor of 2.
5. Therefore, a subset of size 1002 can be chosen such that no two elements are relatively prime.
6. To ensure that any subset of \(\{1, 2, 3, \ldots, 2004\}\) with \( n \) elements contains at least two elements that are relatively prime, \( n \) must be greater than 1002.
7. The smallest such \( n \) is \( 1003 \).

The final answer is \(\boxed{1003}\).