Times Tables and Pascal's Triangle

                       

Pascal invented a triangle of numbers which today we call Pascal’s Triangle. If you go on to study advanced mathematics you will come across this triangle a lot, but today we are going to look at just one or two properties of the triangle.

First you need to draw some of the rows of Pascal’s Triangle and you will need a big piece of paper for this.

You start by putting just the number 1 in the first row and two 1’s in the second row, like this:

                                  

From now on each row begins and ends with a 1, but to get the numbers in the middle, you add the two numbers just above it.

So the second row is 1, followed by 2 (obtained from adding the 1 and 1 in the second row), followed by a final 1. So the triangle now looks like this:

                                      

The next row starts with a 1 as usual and ends with a 1, and the numbers in the middle we get by adding the 1 and the 2 and the 2 and 1, like this:

                                   
I am sure that by now you can see how to get the next row. Begin and end with a 1. Then add the 1 and 3 to get 4; add the 3 and 3 to get 6; add the 3 and the 1 to get 4. This gives us:

You can see that Pascal’s Triangle goes on forever, but I would like you to write down as many rows as you can (which is why you need a large piece of paper). Be very careful to get all the answers correct as if you make a mistake in one row, all the rows below will be wrong too.

Can you see the pattern of the numbers I have ringed in the following diagram? You will need more rows before you can be sure of the pattern.

When I was discussing Pascal’s Triangle with a class of bright 16 year olds many moons ago, one of them said, ‘If the second number in a row is a prime number, it is a factor of all the other numbers in the row, except the ones. If the second number in the row is not a prime number it is not a factor of all the other numbers in the row (although it may be a factor of a few of them).’

I was gobsmacked, but I wondered if it is true, so we drew lots of rows and investigated. Can you work out whether this girl was correct?

In case you didn’t quite understand the question, let me give you a couple of examples.

In the row beginning 1   4   6…, the second number is 4 and this is not a prime number. The girl was therefore saying that it is not a factor of all the other numbers in the row. And, sure enough, 4 is not a factor of 6.

However, in the next row:

              1    5    10    10    5    1

5 is a prime number and it is a factor of all the other numbers in the row (except the 1’s, of course).

So, the question is: Is this always true or is it just true sometimes? To find out you are going to need quite a few rows.

I bet you never thought you would be using prime numbers in this way and prime numbers, as we all know, are not found in the multiplication tables except when you multiply by 1.

Have fun!