Validation of credit card numbers is usually done by Luhn's algorithm:

Example

Passo                                                                                                       Total
Número Original                          :  4   5   5   6   7   3   7   5   8   6   8   9   9   8   5   5   
Tirando o último dígito                  :  4   5   5   6   7   3   7   5   8   6   8   9   9   8   5       
Invertendo                               :  5   8   9   9   8   6   8   5   7   3   7   6   5   5   4       
Multiplicando casas ímpares por 2        :  10  8   18  9   16  6   16  5   14  3   14  6   10  5   8       
Subtraia 9 de todos os números acima de 9:  1   8   9   9   7   6   7   5   5   3   5   6   1   5   8       
Somando todos os números                 :  1   8   9   9   7   6   7   5   5   3   5   6   1   5   8       85
Mod 10: 85, módulo 10 = 5 (último dígito do cartão)

Known Operator Formats

Bandeira                    Intervalo de Início             Número de Dígitos
-----------------------------------------------------------------------------
American Express            34, 37                          15
Diners Club - Carte Blanche 300, 301, 302, 303, 304, 305    14
Diners Club - International 36                              14
Diners Club - USA & Canada  54                              16
Discover                    6011, 622126 até 622925, 644,   16
                            645, 646, 647, 648, 649, 65     
InstaPayment                637, 638, 639                   16
JCB                         3528 até 3589                   16
Laser                       6304, 6706, 6771, 6709          16-19
Maestro                     5018, 5020, 5038, 5893, 6304,   16-19
                            6759, 6761, 6762, 6763          
MasterCard                  51, 52, 53, 54, 55              16-19
Visa                        4                               13-16
Visa Electron               4026, 417500, 4508, 4844, 4913, 16
                            4917    

I pulled it out .

In ASP.NET MVC

By attribute, as answered by @Randrade .

The algorithm used to calculate the Check Digit of Credit Card Numbers is the Luhn's Algorithm , but it is not enough to just validate the information without taking into account the characteristics of a Credit Card number that are:

• Only have numbers
• Have 12 to 19 digits (there is actually no 17 digit CC)
• The first 6 digits should indicate a valid Issuer identification number (IIN). It's called a BIN in banking.
• Passing the Luhnn Algorithm Validation

Verifying the BIN item requires an information base for comparison. There is no logic in the generation of BINs. You can have tracks for example that has jumps and within these jumps the BIN belong to a different institution, including a different country. It is a classic mistake to believe that every card that starts with 4 is VISA and with 5 is MasterCard, for example. Currently in Brazil this is even valid, because we have few Card companies which ends up reserving the 4 and 5 only for these two Flags.

Leaving these aspects aside, as there is no way to validate a BIN via code without some type of query being done, validation can be done with the following algorithm

• Check if the card number is between 12 and 19. Currently in Brazil it is rare to have a card other than 16. Some Hipercard had 19 but are already leaving with 16 as well.
• Verify that the value has only numbers
• Verify that the number passes through the validation of the Luhnn Algorithm

Below a code in C #. It's just a way to understand because there are already in the current versions of ASP.net, Attributes validation for this check, according to @Randrade's answer

The Luhnn Algorithm is represented in answer from @Cigano Morrison Mendez with the proviso that in the last line, where it is

Mod 10: 85, módulo 10 = 5 (último dígito do cartão)

Should be

Mod 10: 85, módulo 10 = 5, Check Digit = 10 - 5 = 5 (último dígito da sequencia)

In fact, there is no need to calculate the check digit . Simply include the term in the sum and compute the module 10 of it. If it is zero it is valid.

    // About the Algorithm
    /**
        @See https://en.wikipedia.org/wiki/Luhn_algorithm
        Steps:
        1 - From the rightmost Digit of a Numeric String, Double the value of every digit on odd positions
        2 - If the obtained value is greather than 9, subtract 9 from it
        3 - Sum all values
        4 - Calculate the Modulus of the value on 10 basis, if is zero so the String has a Luhnn Check Valid
    **/

    public static bool IsValidLuhnn(string val) {

        int currentDigit;
        int valSum = 0;
        int currentProcNum = 0; 

        for (int i = val.Length-1; i >= 0; i--) {           
            //parse to int the current rightmost digit, if fail return false (not-valid id)
            if(!int.TryParse(val.Substring(i,1), out currentDigit))
                return false ;              

            currentProcNum = currentDigit << (1 +i & 1);                                                                    
            //summarize the processed digits
            valSum +=  (currentProcNum > 9 ? currentProcNum - 9 : currentProcNum);

        }  

        // if digits sum is exactly divisible by 10, return true (valid), else false (not-valid)
        // valSum must be greater than zero to avoid validate 0000000...00 value
        return (valSum > 0 && valSum % 10 == 0) ;
    } 


    public static bool isValidCreditCardNumber(string cc) {
        // rule #1, must be only numbers
        if (cc.All(Char.IsDigit) == false) {
            return false;
        }
        // rule #2, must have at least 12 and max of 19 digits
        if (12 > cc.Length || cc.Length > 19) {
            return false;
        }
        // rule #3, must pass Luhnn Algorithm
        return IsValidLuhnn(cc);

    }

Can be tested here