3. (25 points) Write the 90 positive integers $10, 11, \cdots, 99$ on the blackboard, and erase $n$ of them so that the product of all the remaining numbers on the blackboard has a units digit of 1. Find the minimum value of $n$.


3. If the unit digit of the product of all remaining numbers on the blackboard is 1, then all even numbers between $10 \sim 99$ must be erased, and numbers with a unit digit of 5 must also be erased.

Thus, the unit digit of the remaining numbers must be one of $1, 3, 7, 9$.

Notice that the unit digit of $11 \times 13 \times 17 \times 19$ is 9. Similarly, the unit digit of $21 \times 23 \times 27 \times 29, \cdots, 91 \times 93 \times 97 \times 99$ is also 9. Therefore, the unit digit of the product of all these numbers is 9.

Thus, to make the unit digit of the product of all remaining numbers on the blackboard 1, at least one more number should be erased.

Furthermore, by erasing all even numbers between $10 \sim 99$ and numbers with a unit digit of 5, as well as one number with a unit digit of 9, such as 19, the unit digit of the product of the remaining numbers can be 1.

Since there are 45 even numbers between $10 \sim 99$ and 9 numbers with a unit digit of 5, the minimum value of $n$ is
$$
45+9+1=55 \text {. }
$$
(Li Changyong provided)