Question

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. if she made more playlists,. she could make playlists, but this is not the greatest number she could make.

Explanation:

Step1: Find GCD of 32 and 72

To find the greatest number of playlists, we need the greatest common divisor (GCD) of 32 and 72. Prime factorize 32: \(32 = 2^5\). Prime factorize 72: \(72 = 2^3\times3^2\). The GCD is the product of the lowest powers of common prime factors, so \(GCD(32,72)=2^3 = 8\).

Step2: Calculate songs per playlist

For pop songs: \(\frac{32}{8}=4\). For rock songs: \(\frac{72}{8}=9\).

Answer:

The greatest number of playlists Katherine can make is \(8\). Each of these playlists will have \(4\) pop songs and \(9\) rock songs.
(For the other parts, if we consider more playlists, the number would be a factor of 8 less than 8, like 4, 2, 1. For example, if she made more playlists (e.g., 16), she couldn't divide 32 and 72 equally. She could make 4 playlists (not the greatest), etc.)