To find the number of pairs of numbers whose product is 2028 and whose highest common factor (H.C.F.) is 13, we can use the prime factorization method.

1. Prime Factorization of 2028:
To find the prime factorization of 2028, we divide it by its smallest prime factor repeatedly until we get 1.
2028 ÷ 2 = 1014
1014 ÷ 2 = 507
507 ÷ 3 = 169
169 ÷ 13 = 13
13 ÷ 13 = 1

Therefore, the prime factorization of 2028 is 2^2 × 3 × 13^2.

2. Finding the Factors:
To find the factors of 2028, we consider all the possible combinations of the prime factors obtained in the prime factorization.
The factors of 2028 are: 1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 169, 338, 507, 676, 1014, 2028.

3. Pairs of Numbers:
Now, we need to find the pairs of numbers whose product is 2028 and whose H.C.F. is 13.
The pairs of numbers are formed by taking one factor from the list above and dividing it by 13, and then multiplying it by another factor from the list above.

For example, let's consider the factor 13:
13 ÷ 13 = 1

Now, we need to find the other factor that, when multiplied by 1, gives the product 2028.
2028 ÷ 1 = 2028

Therefore, one pair of numbers is (1, 2028).

Similarly, we can find another pair of numbers:
26 ÷ 13 = 2
2028 ÷ 2 = 1014

Therefore, the other pair of numbers is (2, 1014).

Hence, there are two pairs of numbers whose product is 2028 and whose H.C.F. is 13. Therefore, the correct answer is option 'B' (2).