Luhn Check Test: How the Mod 10 Algorithm Works

A Luhn check test runs the Luhn algorithm, also called the mod 10 algorithm, over a string of digits to confirm it passes a simple checksum. When the digit sum is divisible by 10, the number is structurally valid; when the remainder is anything else, at least one digit is wrong or out of place.

The test takes a few lines of code and catches the most common typing mistakes before a number reaches a database. Payment card numbers, IMEI codes, and several national identifier formats all rely on it.

How the Luhn algorithm works

1. Write the digits in order, then count from the rightmost digit, which sits in position 1.
2. Double the value of every digit in an even position (2, 4, 6 and so on).
3. If a doubled result is greater than 9, subtract 9 from it. This is the same as adding the two digits of the result together.
4. Add the doubled digits to the digits you left alone.
5. Divide the total by 10. A remainder of 0 means the number passes.

Worked example

Start with the digits 7 9 9 2 7 3 9 8 7 1 3. Counting from the right, the digits in even positions are 1, 8, 3, 2 and 9. Doubling them gives 2, 16, 6, 4 and 18. After subtracting 9 from 16 and 18, the doubled set becomes 2, 7, 6, 4 and 9.

The digits left alone are 3, 7, 9, 7 and 7. Adding both groups: 2 + 7 + 6 + 4 + 9 + 3 + 7 + 9 + 7 + 7 = 70. Since 70 divides evenly by 10, the string passes the check. Change any single digit or swap two adjacent digits and the total usually stops being a multiple of 10, which is the property the algorithm is designed to exploit.

What a Luhn check test cannot tell you

• Whether an account exists, is open, or holds any balance.
• Whether the number was issued by a real institution.
• Whether the number is active, closed, or reported lost.
• Whether the digits were ever assigned to anyone. Any sequence that satisfies the checksum passes.

Treat the test as a typo filter and a format check, not as evidence about the number itself.

Where the check digit shows up

• Payment card numbers, covered by the ISO/IEC 7812 series.
• IMEI numbers on mobile devices.
• National Provider Identifier numbers in US healthcare.
• Canadian Social Insurance Numbers and some bank account formats.

Testing your own implementation

Known vectors keep your code honest. The string 79927398713 passes, while 79927398710 through 79927398712 and 79927398714 through 79927398719 all fail. Run the same input twice with spaces and hyphens added to confirm your normalization step works, then try an empty string, a string with letters, and an input longer than 32 bits to check for overflow.

Common implementation mistakes

• Counting positions from the left instead of the right.
• Doubling the check digit itself.
• Skipping the subtract 9 step for doubled values.
• Failing to strip spaces, hyphens, and leading or trailing whitespace.
• Storing the running total in a small integer type that overflows on long inputs.