Short answer

The Luhn test is a check digit test. It reads a number from right to left, doubles every second digit, adds those results to the digits it left alone, and checks whether the total divides by 10. A total that ends in 0 means the number passes. Any other total means the number fails.

What the Luhn test checks

The Luhn formula was published by Hans Peter Luhn in 1954. It catches any single mistyped digit and most swaps of two adjacent digits. It is a format check and nothing more. A pass confirms that the digits are internally consistent. It does not confirm that an account exists, that a card is live, that funds are available, or that a device is registered to anyone.

Prerequisites

• The number written as a string of digits, with spaces and dashes removed.
• A calculator, spreadsheet, or short script for the arithmetic.

How to run a Luhn test

1. Write the digits in a row and drop every space, dash, and separator.
2. Start at the rightmost digit and move one position left at a time.
3. Double every second digit you meet, beginning with the second digit from the right.
4. Subtract 9 from each doubled value that is greater than 9.
5. Add the adjusted doubled values and the untouched digits into one total.
6. Divide the total by 10 and look at the remainder. A remainder of 0 means the number passes; any other remainder means it fails.

Worked example

Take the number 79927398713. Reading from the right, the digits in odd positions from the right are 3, 7, 9, 7, 9, 7 and they stay as they are. The digits in even positions from the right are 1, 8, 3, 2, 9, 9 and they get doubled: 2, 16, 6, 4, 18, 18. Values above 9 lose 9, which turns 16 into 7 and each 18 into 9. The adjusted row is 3, 2, 7, 7, 9, 6, 7, 4, 9, 9, 7. Those digits add to 70. Since 70 divides by 10 with no remainder, the number passes.

Generating the check digit

1. Take the payload digits and leave the last position empty for the check digit.
2. Start at the rightmost payload digit and double every second digit moving left.
3. Subtract 9 from every doubled value above 9.
4. Add all the values to get the payload sum.
5. Find the smallest number between 0 and 9 that lifts the payload sum to a multiple of 10.
6. Place that number in the last position as the check digit.

For the payload 7992739871 the values add to 67. The next multiple of 10 is 70, so the check digit is 3 and the full number becomes 79927398713.

Where the Luhn test applies

• Payment card numbers use a Luhn check digit in the final position of the primary account number, as specified in ISO/IEC 7812.
• IMEI numbers on mobile devices carry a Luhn check digit.
• The National Provider Identifier used in United States healthcare carries a Luhn check digit.
• Canadian Social Insurance Numbers and several national identity and account schemes use the same formula.
• Form validation libraries apply the test to catch typos before a number is submitted.

Limits of the test

The Luhn formula has a known blind spot: it misses the transposition of the two-digit sequence 09 and 90, and it misses some double errors. It also says nothing about the rest of a number's structure. A payment card number carries an issuer identification number and a length rule that the Luhn test does not evaluate, and the three or four digit security code printed on a card is not part of the Luhn calculation at all. Two different numbers can pass the test while only one of them is issued to anyone, so a pass is a statement about arithmetic, not about validity in the real world.