Grouping same number of people with least amount of groups

Question

In a political science class, the teacher wants to assign students with the same political affiliation to groups in order to prepare for a debate. She finds that there are 133 Republicans, 190 Democrats, and 171 Independents. If she wants the same number of people in each group and the smallest number of groups, how many total groups will there be?

Answer 1

First, let's list the factors of each of the three numbers:

 

133: 1, 7, 19, 133

 

190: 1, 2, 5, 10, 19, 38, 95, 190

 

171: 1, 3, 9, 19, 57, 171

 

 

 

Notice that all three of these numbers share the greatest common factor of 19. This means that there could be 19 groups. To find how many of each political affiliation should be in each group, divide their number of representatives by 19.

 

133 / 19 = 7

 

190 / 19 = 10

 

171 / 19 = 9

 

 

 

Therefore, there should be 19 total groups, and within each group should be 7 Republicans, 10 Democrats, and 9 Independents.

 

Answer 2

The number of groups is equal to the GCF of 133, 190, 171.

133 = 7 · 19 

190 = 2 · 5 · 19

171 = 3 · 3 · 19

 

The GCF is 19. The teacher should have 19 groups.