What is the property that states that multipling a sum by a number is the same as multiplying each addend in the sum by the number and then adding the products?

The property you're referring to is the Distributive Property. It states that for any numbers (a), (b), and (c), the equation (a(b + c) = ab + ac) holds true. This means that multiplying a sum by a number is equivalent to multiplying each term inside the parentheses by that number and then adding the results together.