To check if a credit card number is valid, you can use the **Luhn algorithm**. This is a simple checksum formula used to identify mistyped or incorrect numbers. Here’s how it works:

==== Steps of the Luhn Algorithm ====

1. **Reverse the order** of the card number's digits.

2. **Double every second digit**, starting from the first digit (which is now the second digit of the reversed number).

3. If any doubled value is greater than 9, **subtract 9 from it** (alternatively, add the digits together; e.g., 12 becomes 1 + 2 = 3).

4. **Sum all the digits** (including both doubled and undoubled digits).

5. **Check if the sum is divisible by 10**. If it is, the card number is valid.

=== Example ===

Let’s check if the following card number is valid: **4539 1488 0343 6467**

1. Reverse it: **7646 3430 8841 9354**

2. Double every second digit: **7, 12, 4, 12, 6, 6, 3, 6, 8, 16, 4, 2, 9, 6, 5, 8**

3. For doubled values greater than 9 (12, 12, 16), add their digits: **7, 3, 4, 3, 6, 6, 3, 6, 8, 7, 4, 2, 9, 6, 5, 8**

4. Sum: **70**

5. Since **70 is divisible by 10**, the card number is **valid**.

=== Quick Check Code in Python ===

Here's a simple Python code to check if a credit card number is valid using the Luhn algorithm:

<sxh python>
def luhn_check(card_number):
    digits = [int(d) for d in str(card_number)][::-1]
    checksum = sum(digits[0::2]) + sum(sum(divmod(2 * d, 10)) for d in digits[1::2])
    return checksum % 10 == 0

# Example usage
card_number = "4539148803436467"
print("Valid" if luhn_check(card_number) else "Invalid")
</sxh>

Here is a C implementation of the algorithm:

<sxh c>
#include <stdio.h>
#include <string.h>
#include <ctype.h>

// Function to validate a Visa card number using the Luhn Algorithm
int isValidVisaCard(const char* cardNumber) {
    int length = strlen(cardNumber);
    int sum = 0;
    int isSecond = 0;

    // Traverse the card number in reverse
    for (int i = length - 1; i >= 0; i--) {
        int digit = cardNumber[i] - '0';

        if (isSecond) {
            // Double every second digit
            digit *= 2;
            if (digit > 9) {
                // Subtract 9 if the doubled value is greater than 9
                digit -= 9;
            }
        }

        // Add digit to the sum
        sum += digit;
        // Toggle the isSecond flag
        isSecond = !isSecond;
    }

    // If the sum is divisible by 10, the card number is valid
    return (sum % 10 == 0);
}

int main() {
    char cardNumber[20];
    printf("Enter the Visa card number: ");
    scanf("%19s", cardNumber);

    if (isValidVisaCard(cardNumber)) {
        printf("The card number is valid.\n");
    } else {
        printf("The card number is invalid.\n");
    }

    return 0;
}
</sxh>

Based on the Luhn algorithm, this code will output whether the credit card number is valid or invalid.

Generating a valid credit card number involves adhering to the structure and validation checks used by credit card companies, primarily the Luhn algorithm. Here’s how to generate a valid credit card number (without real-world use, as actual issuance requires official channels).

=== Steps to Generate a Valid Credit Card Number ===

1. **Choose a Bank Identification Number (BIN)**: The first 6 digits of a credit card (the BIN) identify the issuing bank. For example:
   - VISA typically starts with 4.
   - MasterCard numbers often start with 51-55.
   For this example, we’ll use a generic BIN: `4539 14` (representing a VISA card).

2. **Generate Remaining Digits Except the Checksum**: After the BIN, generate random digits for the rest of the card number, leaving the last digit blank for now. VISA and MasterCard usually have 16 digits, so we’ll generate 9 more digits.

3. **Calculate the Checksum with the Luhn Algorithm**:
   - Apply the Luhn algorithm to calculate the checksum digit (the last digit), which will make the number valid.

=== Example Code to Generate a Valid Card Number ===

Here’s a Python code that generates a valid credit card number:

<sxh python>
import random

def generate_credit_card():
    # Start with a sample BIN (6 digits)
    bin_number = [4, 5, 3, 9, 1, 4]  # Example BIN for VISA
    # Generate the next 9 random digits
    card_number = bin_number + [random.randint(0, 9) for _ in range(9)]
    
    # Calculate the Luhn checksum for the first 15 digits
    def luhn_checksum(card):
        digits = card[::-1]
        checksum = sum(digits[0::2]) + sum(sum(divmod(2 * d, 10)) for d in digits[1::2])
        return checksum % 10

    # Calculate the last digit to make the number valid
    check_digit = (10 - luhn_checksum(card_number)) % 10
    card_number.append(check_digit)

    return ''.join(map(str, card_number))

# Generate a valid credit card number
print(generate_credit_card())
</sxh>