Solution below.

Reveal solution by highlighting:

This one I sort of brute forced. I consider the first integer in my product. It might be 1, 2, 3 etc. Then I consider my product. It might be made of 2 integers, or 3, or 4 etc. In the rows below I consider each beginning integer, and I stop when I hit a 3 digit product. (If I continued, I would hit larger products, and I’m interested in the smallest.) The first column keeps going, until I hit a 3 digit product of 2 integers. (If I continued, I would hit larger products.) I don’t do 1 as a first integer, as the products would basically be the same as those with 2 as the first integer.

Having constructed my table, I look for the smallest 3 digit product. It turns out to be 110, our solution.