Validate credit card numbers, IMEI codes, and other IDs using the Luhn checksum algorithm. Generate check digits with step-by-step visualization.
The **Luhn algorithm** (also called the **modulus 10** or **mod 10** algorithm) is a simple checksum formula used to validate a variety of identification numbers such as credit card numbers, IMEI numbers, Canadian Social Insurance Numbers, and more. Invented by IBM scientist Hans Peter Luhn in 1954, it was designed to protect against accidental errors — not intentional attacks — and remains one of the most widely deployed error-detection algorithms in the world.
Every time you make an online purchase or enter a credit card number, the Luhn algorithm is one of the first checks performed. It catches common transcription errors like single-digit mistakes and adjacent-digit transpositions, ensuring that mistyped numbers are rejected before any network request is sent. The algorithm is computationally trivial, requiring only a single pass through the digits.
Our **Luhn Algorithm Calculator** supports both **validation** (checking whether a given number is Luhn-valid) and **check-digit generation** (computing the digit that, when appended, makes the number valid). The step-by-step table shows every doubling, reduction, and summation so you can see exactly how the algorithm processes each digit — ideal for students, developers, and anyone curious about how card numbers actually work.
Understanding the Luhn algorithm is essential for software developers implementing payment forms, mobile device management systems, or any application that accepts identification numbers. Our calculator provides instant validation with a full step-by-step breakdown, making it a perfect learning tool for computer science students and a quick verification utility for professionals.
Luhn Algorithm: Starting from the rightmost digit (check digit), double every second digit moving left. If doubling produces a value > 9, subtract 9. Sum all digits. The number is valid if the total modulo 10 equals zero. Check digit d = (10 − (sum mod 10)) mod 10.
Result: Valid (sum = 50, 50 mod 10 = 0)
Processing from right to left: alternate digits are doubled. For instance, the 8 in position 2 becomes 16, which reduces to 7 (16 − 9). After doubling and reducing all alternating digits, the total sum is 50, which is evenly divisible by 10, confirming the number is a valid Visa card.
The Luhn algorithm was invented by Hans Peter Luhn, a German-American computer scientist working at IBM, and was described in U.S. Patent 2,950,048 filed on January 6, 1954. Originally designed for use with physical punch-card machines, the algorithm was chosen for its simplicity — it can be computed by hand or with the most basic hardware. Despite being over 70 years old, it remains the standard checksum for credit card numbers and many other identification systems worldwide.
The core insight of the Luhn algorithm is that doubling a single digit and subtracting 9 when the result exceeds 9 is equivalent to adding the two digits of the doubled value. For example, 7 × 2 = 14, and 1 + 4 = 5, which equals 14 − 9 = 5. This digit-sum shortcut means the entire algorithm can be performed with only addition and comparison, no division required — a critical advantage in the 1950s when division was expensive on early computers.
While the Luhn algorithm catches all single-digit substitution errors and all transpositions of adjacent digits, it fails to detect some transpositions of non-adjacent digits and cannot detect certain twin-error patterns. For applications requiring stronger error detection, the **Verhoeff algorithm** (based on the dihedral group D₅) catches all single-digit errors and all transposition errors. The **Damm algorithm** offers similar strength with a simpler implementation. However, for the vast majority of use cases — especially credit card validation where the error model is primarily accidental human typos — Luhn remains the industry standard due to its simplicity and speed.
It verifies that a number has not been mistyped by computing a checksum. The last digit (check digit) is chosen so that the sum of all processed digits is divisible by 10. It catches about 96% of single-digit errors and all single transpositions of adjacent digits.
No. The Luhn algorithm only detects accidental errors such as typos. It provides no security against intentional manipulation. Generating a Luhn-valid number is trivial — actual fraud protection relies on additional systems like CVV, 3D Secure, and bank authorization.
Credit and debit card numbers (Visa, Mastercard, Amex, Discover, JCB), IMEI numbers for mobile phones, Canadian Social Insurance Numbers (SIN), some National Provider Identifier (NPI) numbers in healthcare, and many other ID systems. Use this as a practical reminder before finalizing the result.
Because the final validity check is whether the digit sum modulo 10 equals zero. The name "mod 10" (or "modulus 10") describes this final step, though the algorithm also involves doubling and reduction steps.
It misses some transpositions of non-adjacent digits and most cases where two or more errors cancel each other out. For example, swapping 09 and 90 will not be caught. For stronger error detection, more complex algorithms like the Verhoeff algorithm are used.
Append a zero in place of the check digit, run the Luhn algorithm, and compute (10 − sum mod 10) mod 10. The result is the check digit that, when placed at the end, makes the full number Luhn-valid.