Question

The product of two numbers is the same as one of the factors. How can you use properties of multiplication to determine what the the factors are?

Answer 1

I see two possibilities here:
either the Idendity property is at play and one of the numbers is 1 (and the product is equal to the other number)

or one of the two numbers is 0 and the product is 0 too. (multiplication by 0).

Answer 2

The properties of multiplication are:

 1) Commutative property: When two numbers are multiplied, the product is the same regardless of the order of the multiplicands

 2) Associative property: When three or more numbers are multiplied, the product is the same no matter how the factors are grouped.

 3) Property of neutral element: The product of any number by one is the same number.

 4) Distributive property. The sum of two numbers by a third is equal to the sum of each summand by the third number.

 5) Any number multiplied by zero is equal to zero.

 Using properties 3 and 5 we have two different results:
 $(a) * (1) = a (a) * (0) = 0$
 Where,
 a: real number other than zero.

 Answer:
 Using properties of multiplication the factors are:
 0 and 1