Look on the back of any modern book and you will find an ISBN code. These are used to identify every book that is now published. Before January 1st 2007, they were 10 digits long but books published since 2007 have codes which are 13 digits long. This investigation only works for those published before 2007.

For instance one of Tolkein's books has an ISBN code on the back of $$0-04-823208-4.$$

Take this code and calculate this sum: $$(1\times 0)+(2\times 0)+(3\times 4)+(4\times 8)+(5\times 2)+(6\times 3)+(7\times 2)+ $$ $$+(8\times 0)+(9\times 8)+(10\times 4) $$

Now find a small stack of books and repeat this sum for each of their ISBN codes. Sometimes the last number of an ISBN code is replaced with an X. If it is count this as 10.

Can you see what the answers always have in common?

In the ISBN code below there is one number that has been replaced with a question mark.

Do you know what this number is for it to be a valid ISBN code?$$0-74755?-19-1$$

What do you think is the purpose of this property of ISBN codes?