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Check a card number against the Luhn checksum and detect its brand. Free, runs entirely in your browser — no signup, and nothing you paste in is uploaded.
The Luhn checksum doubles every second digit from the right, subtracts nine from any result above nine, and requires the total to be divisible by ten. It was designed to catch the two mistakes humans actually make — a single mistyped digit, and two adjacent digits swapped — and it catches almost all of both.
What it does not do is tell you a card exists. A number can pass Luhn and belong to no account at all, which is exactly why it belongs in a form as instant feedback on a typo and never as a substitute for authorisation. Brand detection is separate again: it is pattern matching on the leading digits and length, not part of the checksum.