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Credit card check digits work through an algorithm called the Luhn algorithm, which helps detect errors in card numbers. Here’s how it works:

1. Starting from the rightmost digit (excluding the check digit), double every second digit.
2. If doubling results in a number greater than 9, subtract 9 from it.
3. Add up all the digits (both doubled and untouched).
4. The total should be a multiple of 10. If it's not, the card number is invalid.

The check digit ensures the card number is mathematically consistent, helping catch common entry errors.
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The Luhn algorithm and check digits are necessary for several reasons:

1. Error detection: They help identify simple mistakes, such as mistyping a digit or swapping numbers, which are common during manual entry.
2. Fraud prevention: The algorithm makes it harder for attackers to randomly guess valid credit card numbers, since invalid numbers won't pass the Luhn check.
3. Efficient processing: The algorithm enables quick validation of card numbers at the point of entry, reducing the chances of processing incorrect or fraudulent numbers.