In a digit 13 ISBN number, can you think of errors that the check digit system cannot identify and give an example to explain why this is the case?

We'll put that right now and show the calculation for the correct number, and for both of those numbers that I entered with errors.

This is the process:

Take the first 12 digits of the 13-digit ISBN

Multiply each number in turn, from left to right by a number. The first digit is multiplied by 1, the second by 3, the third by 1 gain, the fourth by 3 again, and so on to the eleventh which is multiplied by 1 and the twelfth by 3.

Add all of the 12 answers.

Do a modulo 10 division on the result from step 2. (Don't know what a modulo 10 division is? It's easy. It's just the remainder from a whole number division by 10. I bet you learned to do that in junior school, before you even learned about decimal fractions.)

Take that remainder result from step 4.If it's a zero, then the check digit is zero. If the remainders isn't zero then subtract the remainder from 10. The answer to that is your check digit.