RECTANGLES AND PRIMES

Arranging objects in rectangular arrays--an overview

Four objects can be arranged in a column, like this:

or in a row, like this:

 With more columns, they can also be arranged in three rectangular arrays, like this: For six objects there are four different rectangular arrays:

 With more objects you might expect more rectangular arrays. How many rectangles can you form using seven objects? What about five objects? Or eight, nine, or ten? Try entering these numbers in the box at the bottom right in the applet below: (Move cursor into box, click, type number, hit ENTER) Now enter some bigger numbers. Some will form groups of rectangles that flow off the edge of the box. Even so, it is often possible to decide how many rectangular arrays have been formed.

 Prime and composite numbers As you experiment with various numbers, you will discover that for some numbers, like seven, there are only two rectangular arrays. These numbers are called prime numbers. For others, like six, there are more than two. These are called composite numbers. You can figure out if a given whole number greater than one is prime or composite by arranging that number of objects into all possible rectangular arrays. The applet above does just that! Use the applet to find all primes less than: 30; 50; 100.

# The number 1 is a special case--

it is neither prime nor composite, because there is only one rectangular array:

1 Column

1 Row

# Transposed Arrays

For any rectangular array, for example, this one:

2 Columns

1 Row

there is always another array with rows and columns interchanged, like this:

3 Columns

2 Rows

These two arrays are said to be transposed.

Here are more examples:

1 Column

5 Rows

5 Columns

1 Row

array
transposed array

3 Columns

4 Rows

4 Columns

3 Rows

array
transposed array

6 Columns

6 Rows

6 Columns

6 Rows

array
transposed array

 For each array with m rows and n columns there will always be a transposed array with n rows and m columns. This will be a new array if m and n are different, but will be the same array if m and n are equal. So, when we look for all possible rectangular arrays of a given set of objects, it suffices to find those in which the number of rows is no larger than the number of columns.