If you take two integers and look at the difference between the
square of each value, there is a nice relationship between the
original numbers and that difference.
The Excel file Difference of Two
Squares.xls lets you explore this.( Right-click on the link,
"Save Target As", and select where you want the file to be
Sometimes a table of values helps, and the Excel file As a Table.xls gives you
Look at the numbers and see if you spot anything that might be
For example under the 6 you'll see 35, 32, 27 and 20 (notice
If you know about the Difference of Two Squares already then
move straight on to Beyond Squares.
In the file "As a Table", the formula in D3 is: = ABS ( $C3 ^ 2
- D$2 ^ 2 ).
This calculates the (positive) difference between C3 squared and
The dollar sign on the C of C3 means
that if this formula is copied elsewhere the C part of the cell
reference will not be adjusted for the new position. The formula
will always pick up its first value from column C. In the same way
the dollar sign on the 2 of D2 means that
when this formula is copied to new positions the cell reference
will keep with row 2.
Once this formula has been entered in D3 , right-click on D3 and
choose Copy, select the whole table including D3, right-click and
You may have discovered that when the original numbers differ by
2 the difference of their squares always contains a factor of 2,
but when the original numbers differ by 3, the difference of their
squares contains a factor of 3, and so on. Of course the important
thing to do when you reach that point is to understand why this is
After that, a natural extension to this investigation would be
to try other powers. When the original numbers differ by 2 does the
difference of the cubes have 2 as a factor? Is the difference of
fourth powers also a multiple of 2 ? And so on.
In general : is an - bn always a multiple
of a - b for all positive integer values of n?
Here's an Excel sheet to give you some useful results as you
think about this (with a spinner to set the power from 2 to 6)
Table of Powers.xls
The cell D3 contains a very similar formula to that used in the
"As aTable" file earlier but the power is the value in cell P3
instead of plain 2.
We don't really want to check each number in the table to see
whether it's a multiple of the difference between its two source
Conditional Formatting is perfect for this job. Excel does the
check for us and uses a colour change to indicate the outcome of
Conditional Formatting using a formula : Select D3 and choose
Conditional Formatting from the Format drop-down menu.
You will see the following formula = MOD ( D3, ABS ( D$2-$C3 ) )
This tests to see if the cell, D3, when divided by the
difference between D2 and C3, gives a remainder of zero.
In other words, it tests whether D3 is a multiple of ABS ( D2 -
C3 ). The font colour changes from blue to red when this condition
is satisfied - it seems that it always is.
So it looks like an - bn , where n is a
positive integer, is always a multiple of a - b , but can we see
This is not easy to do, but is a very interesting result.
Perhaps try explaining this for n = 3 , then n = 4 , and so on.
If you do manage that, you are just one step away from seeing what
the other factor must be. For the difference of two squares the
factors were (a - b) and (a + b) For the difference of two cubes
the factors are (a - b) and ( ? ? ? ) For the difference of two
fourth powers the factors are (a - b) and ( ? ? ? ? ). What is the
If you get this far you really are doing some serious