1. How many pairs of numbers $(a, b)$ with $a<b$ can you choose from the set $1,2,3, \ldots, 2005$ such that $a+b$ is a multiple of 5?
1. If $a+b$ is a multiple of 5, then one of the following cases must be true:

a) both numbers are multiples of 5, or

b) one number is a multiple of 5 +1 and the other is a multiple of 5 +4, or

c) one number is a multiple of 5 +2 and the other is a multiple of 5 +3.

Among the 2005 numbers, there are 401 multiples of 5, multiples of 5 +1, multiples of 5 +2, multiples of 5 +3, and multiples of 5 +4.

In case a) there are $\frac{401 \times 400}{2}$ possibilities,

in case b) $401 \times 401$ and in case c) also $401 \times 401$.

In total, therefore, $80200+160801+160801=401802$.