“To Err is Human…” so goes the old truism, but luckily computers can often help in error correction. As anyone who has typed a long sequence of numbers (such as a credit card number) knows, it is extremely easy to make a mistake while typing. Common errors include typing the wrong number, switching two numbers, forgetting a number, and inserting an extra number.

There are many different error detection methods, both for detecting errors in binary numbers and in decimals. One well-designed and widely used check method for decimal numbers is called the IBM Check.

The check is actually quite simple. Given an eight digit number, the number is valid if:

(f(d1, 2) + d2 + f(d3, 2) + d4 + f(d5, 2) + d6 + f(d7, 2) +  c) mod 10 = 0

where c is the check digit (d₈, the last digit in the sequence) and f(n) is defined as:

public int f(m, n) {
           char[] tmp = Integer.toString(m * n).toCharArray();

           int val = 0;

           for (int i=0; i < tmp.length; i++) {
           	  val += Character.getNumericValue(tmp[i]);
           }

           return val;
}

So f(m, n) simply multiplies the two numbers and adds the digits of the product. To illustrate:

f(7, 2) --> 7*2 = 14 --> 1+4 = 5

So f(7, 2) returns 5. Now that the check process is clear, it is quite easy to generate a valid sequence. To generate an eight digit sequence, seven random digits would be plugged in the equation, leaving c as the only unknown. Simple algebra and logic is all that is then required to find c. And then voilà! — an IBM check-valid number.

It should be noted that the IBM check has a few weaknesses. Like any other check process, it is possible that a user might enter in a wrong number that still satisfies the check. Furthermore, flipping a 9 and a 0 won’t be detected by the check because f(9, 2) = 9 and f(0, 2) = 0. The IBM check is still, however, a very useful tool and a very popular one. In fact, credit cards often use this check. Think of that next time you type those sixteen digits.

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