Suppose we use the digits 2, 3, 5, 6, and 8 once each to make a three-digit number and a two-digit number, and then we multiply them together. For example, we could do 235 ×68, giving us a product of 15,980. YOU MUST USE EACH DIGIT ONCE. For example, 865 * 86 would not work because both numbers use 8 and 6 more than once.

(a) Find the largest possible product and explain why you don’t think you can make anything larger.

(b) Find the smallest possible product and explain why you don’t think you can make anything smaller.

(c) Repeat parts (a) and (b) with the digits 1, 4, 5, 7, and 9. Remember, you must use each digit one time only.

(d) Make a conjecture for how to solve this problem given any five distinct digits, based on your answers to parts (a), (b), and (c). Explain the reasoning that led to your conjecture, and test it out on a new set of five digits to see if you were right.