Find two prime numbers that between them use each of the four digits 1, 2, 3 and 5 exactly once. For example, you could make 53 and 21, however, 21 = 3 7 is composite (non-prime).
As no 2-digit prime ends in 5, there can be no 2-digit solution. The only combinations would be: 21 (3 7) + 53 and 23 + 51 (3 17), in which case at least one number in the pair is composite.
So the only possibility is that the solution comprises a single digit prime and a 3-digit prime.
The single digit cannot be 2 or 5, as the sum of the remaining digits 135 and 123, respectively, are divisible by 3; hence any combination of them would be divisible by 3.
Therefore, the single digit prime must be 3 and the 3-digit prime could be either 251 or 521.