Explanation:

Let the numbers be 13x and 13y  

 (? HCF of the numbers = 13)
13x × 13y = 2028
=> xy = 12
co-primes with product 12 are (1, 12) and (3, 4)  

(? we need to take only  co-primes with product 12. If we take two numbers with product 12,

but not co-prime, the HCF will not remain as 13)
Hence the numbers with HCF 13 and product 2028
= (13 × 1, 13 × 12) and (13 × 3, 13 × 4)
= (13, 156) and (39, 52)
Given that the numbers are 2 digit numbers
Hence numbers are 39 and 52

Hence, option 2 is the correct answer.