Question

1. the glee club has 120 cupcakes to sell. they have decided to arrange the cupcakes in the shape of a rectangle so that each row has an even number of cupcakes and each column has an odd number of cupcakes. how many arrangements of cupcakes can they create? explain.

Explanation:

Step1: Factorize 120

$120 = 2^{3}\times3\times5$
Let the number of rows be $r$ (even) and the number of columns be $c$ (odd). Then $120 = r\times c$. Since $r$ is even, it must have a factor of 2. Since $c$ is odd, it has no factor of 2.

Step2: Express 120 as product of even - odd

The odd - factors of 120 are the factors of $3\times5 = 15$. The factors of 15 are 1, 3, 5, 15.
For each odd - factor $c$ of 15, we can find an even number $r=\frac{120}{c}$ such that $r\times c=120$.
When $c = 1$, $r = 120$; when $c=3$, $r = 40$; when $c = 5$, $r=24$; when $c = 15$, $r = 8$.

Answer:

4