Readings for Session 12 – Factors, Multiples, and Divisors

 

How does the following problem relate to the factors in a product?

Carla has twelve gum balls and wants to share them among friends where each person receives the same number of gumballs.  Carla has several choices for how she shares her gumballs depending on how many friends she shares her gumballs with. What are all the possible ways she can share her gum balls —number of people and number of gumballs each receives?

        Since each person must receive a whole gumball, the problem is asking for all the possible natural number products that can be formed where the product is twelve. That is,

1 × 12, 2 × 6, 3 × 4, 4 × 3, 6 × 2, and 12 × 1.

The possibilities are:

She keeps all twelve gumballs, 1(12).

She and a friend each get six gumballs, 2(6).

She and two friends each get four gumballs, 3(4).

She and three friends each get three gumballs, 4(3).

She and five friends each get two gumballs, 6(2).

She and eleven friends each get one gumball, 12(1).

        We may consider the above problem in three different ways: What are all the ways two natural number factors give a product of twelve? What are all the ways we can multiply two natural numbers to get twelve? What are the possible natural number divisors of twelve that give a natural number quotient? These different perspectives for the above problem motivate the concepts of factors, multiples, and divisors.

  

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