## Task 2 - 290912

Is there a prime number among all five-digit numbers that can be written using exactly the digits 0, 1, 2, 3, 4?
Answer: Yes, for example 10243.

A justification for this answer is still missing. Additionally, some introductory remarks are given, which are not necessarily part of a complete solution to the problem:

If a prime number has the mentioned properties, then none of the digits 0, 2, 4 can be its unit digit. The effort required to check whether a number is prime is smaller the smaller the number to be checked is. Therefore, it is practical to start with the smallest five-digit number that can be written with the mentioned digits and has 1 or 3 as its unit digit, i.e., with the number 10243.

This number is proven to be a prime if it is not divisible by any prime number smaller than 10243. Since $\sqrt{10243}<102$, it suffices to check only the prime numbers $\leq 101$ as divisors. Because if 10243 has a divisor $\geq 102$, then in the factorization of 10243, it must appear together with a factor that is smaller than 102, whose prime factors have already been checked.

Using a table of prime numbers and the SR 1 (or other computational aids), it is quickly established that none of the prime numbers 2, 3, ..., 101 is a divisor of 10243. This justifies the answer.

Remark: All prime numbers with the mentioned properties are: 10243, 12043, 20143, 20341, 20431, 23041, 24103, 30241, 32401, 40123, 40213, 40231, 41023, 41203, 42013, 43201.