A Luhn algorithm test is a checksum check that confirms whether a string of digits is internally consistent with its final check digit. The test doubles every second digit from the right, reduces any result above 9, adds everything together, and expects the total to divide by 10 with no remainder. Passing the test means the number is well formed, not that it belongs to a real card, account, or device.

How the Luhn algorithm test works

The formula is also called mod 10. Hans Peter Luhn, an IBM researcher, developed it in the 1950s and patented it in 1960. The procedure runs the same way on any length of number:

1. Start at the rightmost digit, which is the check digit, and leave it as it is.
2. Move left one digit at a time. Double the value of every second digit.
3. If a doubled value is greater than 9, subtract 9 from it (the same result as adding its two digits).
4. Add all the digits from steps 1 to 3 together.
5. Divide the total by 10. A remainder of 0 means the number passes.

A worked example

Take the sequence 4539 1488 0343 6467. The last digit, 7, is the check digit. Doubling every second digit from the right gives 6 to 12, 6 to 12, 4 to 8, 0 to 0, 8 to 16, 1 to 2, 3 to 6, and 4 to 8. After reducing the values above 9, the doubled column sums to 37.

The digits that were not doubled are 4, 3, 3, 8, 4, 9, and 5, which add to 36. Adding the untouched check digit of 7 gives 37 + 36 + 7 = 80. Since 80 divides by 10, this sequence passes the Luhn test.

What the test proves and what it does not

The check digit exists to catch data entry mistakes before a number travels further into a system. Its coverage is specific:

• It catches any single mistyped digit.
• It catches most swaps of two adjacent digits, with 0 and 9 as a known blind spot.
• It does not confirm that an account exists, is open, or has funds.
• It does not verify a card's expiration date, security code, or billing address.
• It says nothing about authorization, which only the issuing institution can grant.

Because the arithmetic is public, producing digits that satisfy the test takes seconds. That is exactly why a passing result is treated as a formatting check and never as proof of validity.

Where the Luhn check digit appears

Payment card numbering standards place the Luhn check digit at the end of a primary account number, so card numbers, and many other identifiers, carry one:

• Credit, debit, and prepaid card numbers
• IMEI numbers on mobile devices
• National Provider Identifiers used by US healthcare providers
• Canadian Social Insurance Numbers
• SIM card serial numbers and various national ID numbers

Common mistakes when running a Luhn check

• Doubling from the left instead of the right, which shifts the whole pattern.
• Leaving spaces, hyphens, or letters in the input before counting digits.
• Forgetting to reduce doubled values above 9, which inflates the total.
• Reading a pass as a confirmation that the number is live or assigned.

Strip the input to digits, count from the right, reduce properly, and compare the remainder to zero. That is the entire test.