Katherine has 32 pop songs and 72 rock songs on her mp3 player that she will be organizing into playlists. She wants to put an equal number of pop songs and an equal number of rock songs into each playlist. Each song will go into only one playlist. Use the drop-down menus to complete the statements below about the number of playlists Katherine can make.

The greatest number of playlists Katherine can make is
__. Each of these playlists will have __
pop songs and __
rock songs.

Katherine will make a total of 8 playlists. Each playlist will contain 4 pop songs and 9 rock songs.

Step-by-step explanation:

The greatest common factor (GCF) is the greatest integer, common to all, that allows to divide them exactly to all. Or put another way, it is the greatest common divisor to all those numbers.

One way to calculate this value is to obtain all the numbers that are raised, their divisors. The highest divisor that is repeated in all numbers is the Greatest Common Factor.

To be able to distribute the 32 pop songs and 72 rock songs in smaller playlists with the same number of songs of each genre, you must choose a number that is 32 and 72 divisor. As the capacity of all playlists has to be the same , you choose between the divisors of 32 and those of 72 (common divisor). That is, you are calculating the greatest common factor.

You should keep in mind that you must choose the common divisor that is maximum so that the playlists contain as many songs as possible.

The divisors of 32 are 1, 2, 4, 8, 16 and 32.

The divisors of 1, 2, 3, 4, 6, 8, 9,12, 18, 24, 36 and 72.

You can see that the GCF is 8. This means that  Katherine will make a total of 8 playlists.

To obtain the quantity of songs of each genre you divide the total of each genre by that factor in common:

Pop songs 32÷8=4

Rock songs 72÷8=9

Then, each playlist will contain 4 pop songs and 9 rock songs.

Another way to find the GCF is to first find the prime factors of each number:

32 = 2 * 2 * 2 * 2 * 2

72 = 2 * 2 * 2 * 3 * 3

Then, you identify the prime factors that both numbers have in common and multiply them. In this case, 2 is a prime factor that appears in both numbers, and that appears multiplied 3 times. So  the GCF is 2*2*2, or 2³, which results in 8.