Mathematics of Credit Card Numbers

Most credit cards have either a 16 digit number (Master Card and Visa) or a 15 digit number (American Express). Some cards have 13 digit numbers but most of the cards have 15 or 16 digits.
 
The breakup of credit card digits is as follows:
• The first 6 digits of the credit card number represent the Credit Card issuer.
• From the 7th digit till the second-last digit (not counting the last digit) is the credit card number of the person who owns the credit card.
• The last digit is called the “check sum” digit. It is a way to see if the credit card number is correct or not.

What this means is that the last digit in a valid credit card number depends on the other digits in the credit card number.
 
How do we know if the credit card number is valid or not?
• Starting from the right most digit, multiply each digit alternatively by 1 and by 2. So the rightmost digit by 1, the next digit by2, the next by 1 and so on till you reach the leftmost digit.
• If any of the multiples is more than 9, then subtract 9 from it
• Once you complete the steps above, add up all the numbers you get
 
If the total you get above is a multiple of 10, then the credit card number is valid, otherwise it is not
Let’s do an example:
We take an imaginary credit card number: 4328 7723 1054 6628
Is this credit card number valid or not?
 
Digits 4 3 2 8 7 7 2 3 1 0 5 4 6 6 2 8
Multiply by 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
Result 8 3 4 8 14 7 4 3 2 0 10 4 12 6 4 8
If >9, subtract 9 8 3 4 8 5 7 4 3 2 0 1 4 3 6 4 8
 
Add tte numbers in the last row:
8 + 3 + 4 + 8 + 5 + 7 + 4 + 3 + 2 + 0 + 1 + 4 + 3 + 6 + 4 + 8 = 70
Since 70 is a multiple of 10 (7 X 10 = 7), therefore the card number is a valid credit card number.
Pick up any credit card of your parents (or anyone else in your home) and try this out.
 
If we know all the digits of a credit card except the last (check) digit, can you calculate it? Yes.
Using the same logic as above. We know that in a valid credit card, the net sum has to be a multiple of 10.
Let’s look at another credit card number without the last digit.
 
4109 0092 3028 740X
What is X?
We will do the same table as above:
Digits 4 1 0 9 0 0 9 2 3 0 2 8 7 4 0 X
Multiply by 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
Result 8 1 0 9 2 0 18 2 6 0 4 8 14 4 0
If >9, remove 9 8 1 0 9 2 0 9 2 6 0 4 8 5 4 0
 
Add the numbers in the last row:
 
8 + 1 + 0 + 9 + 2 + 0 + 9 + 2 + 6 + 0 + 4 + 8 + 5 + 4 + 0 = 58
Now we know that in a valid credit card, this number should be a multiple of 10. The closest multiple of 10 to 58 is 60. And to reach a total of 60, we need to add 2. So the check digit should be 2.
 
Therefore the last digit of the credit card is 2 and the full credit card number is:
4109 0092 3028 7402
What are the advantages of using the check digit?
There are two main advantages:
It allows people from just generating credit card numbers. There is a logic to generating these numbers.
When people input card numbers (say on the internet), a check digit allows the computer system to do a quick check to see if the number is correct before it goes for processing and a wrong card number gets rejected