RECTANGLES AND PRIMES |

Arranging objects in rectangular arrays--an overview

Fourobjects can be arranged in a column, like this:

or in a row, like this:

With more objects you might

expectmore rectangular arrays. How many rectangles can you form usingsevenobjects?What about

fiveobjects? Oreight,nine, orten?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.

Primeand composite numbersAs 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

is a special case--1it 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

twoarrays are said to betransposed.

Here are more examples:

1 Column

5 Rows

5 Columns

1 Row

arraytransposed array

3 Columns

4 Rows

4 Columns

3 Rows

arraytransposed array

6 Columns

6 Rows

6 Columns

6 Rows

arraytransposed array

For each array with rows and m
columns there will always be a transposed array with n rows
and n columns. This will be a new array if m
andm are different, but will be the same array if n
and m 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.n |