Product Pairs


All natural numbers can be written as the product of two natural numbers. Prime numbers can be written as the product of 1 and themselves. Composites can be written in more than one way as a product of two natural numbers. For instance,

12 = 1 \cdot 12 = 2 \cdot 6 = 3 \cdot 4.

So 12 can be written as the product of two natural numbers in 3 distinct ways.
(Note: We consider a \cdot b and b \cdot a to be the same product pair.)
What is the smallest odd number which can be factored into the product of two natural numbers in exactly 40 di fferent ways? And what about the smallest even number?

(Solution)