# A theater can seat 648 people. The number of rows is less than the number of seats in each row. How many rows of seats are there?

Dec 9, 2017

If you want the number of rows and number of seats in each row to be as equal as possible (with rows less than seats in each row and assuming the same number of seats in each row, then
$\textcolor{w h i t e}{\text{XXX}} 24$ rows

#### Explanation:

There are many possible arrangements (assuming the same number of seats in each row; and even more if you can have different numbers).

{: (ul("rows"),color(white)("xxx"),ul("seats per row")), (1,,648), (2,,324), (3,,216), (4,,162), (6,,108), (...,,....) :}
$\textcolor{w h i t e}{\text{XXX}}$and so on.....

If we limit ourselves to cases where the number of rows and the number of seats per row are close:
$\textcolor{w h i t e}{\text{XXX}} {26}^{2} = 625 < \textcolor{b l u e}{648} < 676 = {26}^{2}$
So we are looking for factors as close as possible to 25 and 26#

We can factor
$648 = \underbrace{2 \times 2 \times 2} \times \underbrace{3 \times 3} \times \textcolor{g r e e n}{3} \times \textcolor{m a \ge n t a}{3}$
$\textcolor{w h i t e}{\text{xxxxxxx")=8color(white)("xxxxx}} = 9$
So a reasonable bi-factoring with factors close in size would be
$\textcolor{w h i t e}{\text{XXX}} 8 \times \textcolor{g r e e n}{3}$ and $9 \times \textcolor{m a \ge n t a}{3}$
$\textcolor{w h i t e}{\text{XXX}} = 24$ and $27$

With the number of rows less than the number of seats per row,
this gives:
$\textcolor{w h i t e}{\text{XXX}} 24$ rows