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The Luhn algorithm or Luhn formula, also known as the "modulus 10" or "mod 10" algorithm, named after its creator, IBM scientist Hans Peter Luhn, is a simple check digit formula used to validate a variety of identification numbers.
To calculate the check digit, take the remainder of (53 / 10), which is also known as (53 modulo 10), and if not 0, subtract from 10. Therefore, the check digit value is 7. i.e. (53 / 10) = 5 remainder 3; 10 - 3 = 7. Another example: to calculate the check digit for the following food item "01010101010x". Add the odd number digits: 0+0+0+0+0+0 = 0.
The total sum of digits is 14 (0 + 2 + 2 + 1 + 4 + 5). The number that must be added to obtain the next multiple of 6 (in this case, 18) is 4. This is the resulting check code-point. The associated check character is e.
There are two modulo 11 algorithms which use different repeated weighting factor patterns: the IBM algorithm which uses (2,3,4,5,6,7), and the NCR algorithm which uses (2,3,4,5,6,7,8,9). Get the sum of the string by looping through each character and multiply it by a weight from 2 to 7 (IBM) or 2 to 9 (NCR) depending on its position.
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, called the modulus of the operation. Given two positive numbers a and n, a modulo n (often abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor. [1]
Virtually all programming languages have a modulo operator for step 3; some languages (also) return the decimal portion which must be multiplied by 11 to get a proper modulo. 11 is used as divisor because a container number has 11 letters and digits in total. In step 1 the numbers 11, 22 and 33 are left out as they are multiples of the divisor.
Not to worry — here’s a quick breakdown of how to write numbers in words on a check. Check Out: 3 Things You Must Do When Your Savings Reach $50,000. Rules for Writing Numbers in Words on a Check.
Primitive root modulo m: A number g is a primitive root modulo m if, for every integer a coprime to m, there is an integer k such that g k ≡ a (mod m). A primitive root modulo m exists if and only if m is equal to 2, 4, p k or 2p k, where p is an odd prime number and k is a positive integer.