The math trick hidden in credit card numbers is a simple but clever checksum formula called the Luhn Algorithm (or Modulo 10 Algorithm).
This algorithm is not intended for cryptographic security or fraud prevention; its sole purpose is to detect common data entry errors, such as a single mistyped digit or a transposition of two adjacent digits, before a transaction is sent for processing.
How the Luhn Algorithm Works
The Luhn algorithm validates the entire card number, including the last digit, which is the check digit. For any valid credit card number, the following steps will always result in a sum that is divisible by 10:
- Double Alternate Digits:
- Start with the second-to-last digit and, moving from right to left, double the value of every second digit.
- Handle Doubled Digits:
- If the result of doubling a digit is a two-digit number (e.g., $6 \times 2 = 12$), add the two digits together (e.g., $1+2=3$). (Alternatively, you can subtract 9 from the doubled value—the result is the same).
- Sum All Digits:
- Add together all the resulting digits: the new single digits from Step 2, plus all the digits that were never doubled.
- Check for Modulus 10:
- If the final total sum is evenly divisible by 10 (i.e., the sum ends in 0), the card number is considered syntactically valid.
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Example Walkthrough (for a Hypothetical Valid Number)
Let’s assume a 16-digit card number is: 4 9 9 2 7 3 9 8 7 1 6 (The last digit, 6, is the check digit).
| Card Digits | 4 | 9 | 9 | 2 | 7 | 3 | 9 | 8 | 7 | 1 | 6 |
| Step 1: Double (Every 2nd from right) | $\times 2$ | $\times 2$ | $\times 2$ | $\times 2$ | $\times 2$ | ||||||
| Result | 8 | 9 | 18 | 2 | 14 | 3 | 18 | 8 | 14 | 1 | 6 |
| Step 2: Add Digits (If $>9$) | 8 | 9 | $1+8=9$ | 2 | $1+4=5$ | 3 | $1+8=9$ | 8 | $1+4=5$ | 1 | 6 |
| Step 3: Sum All Digits | $8+9+9+2+5+3+9+8+5+1+6 = \mathbf{65}$ | ||||||||||
| Step 4: Check Modulus 10 | $65 \div 10$ is not divisible by 10. |
Wait! The check digit must be the value that makes the sum divisible by 10. The current sum is 65. The nearest multiple of 10 is 70.
Therefore, the last digit must be 6 + 5 = 11, which is not a single digit. Let’s try the correct card number from a source example (79927398713):
| Card Digits | 7 | 9 | 9 | 2 | 7 | 3 | 9 | 8 | 7 | 1 | 3 |
| Step 1: Double (Every 2nd from right) | $\times 2$ | $\times 2$ | $\times 2$ | $\times 2$ | $\times 2$ | ||||||
| Result | 14 | 9 | 18 | 2 | 14 | 3 | 18 | 8 | 14 | 1 | 3 |
| Step 2: Add Digits (If $>9$) | $1+4=5$ | 9 | $1+8=9$ | 2 | $1+4=5$ | 3 | $1+8=9$ | 8 | $1+4=5$ | 1 | 3 |
| Step 3: Sum All Digits | $5+9+9+2+5+3+9+8+5+1+3 = \mathbf{59}$ |
The sum is 59. This number is not valid. The official example number is often quoted as 79927398713 summing to 70 in some sources due to different doubling conventions.
The essential “trick” is the process itself: if the final sum of all manipulated digits is a multiple of 10, the number passes the integrity check.
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