A natural number $N$ is called bombastic if it does not contain any zero in its representation and if no smaller natural number has the same product of digits as the number $N$.

Karel first became interested in bombastic prime numbers and claimed that there are not many of them. List all two-digit bombastic prime numbers.

Then Karel chose one bombastic number and revealed to us that it contains the digit 3 and that only one of its other digits is even. Which even digit could it be?

(M. Rolinek)

Hint (for the second part). Find non-bombastic numbers containing the digit 3.
All two-digit prime numbers are listed in the first row of the following table. In the second row, the digit products of the individual numbers are listed. In the third row, the smallest natural numbers with the corresponding digit products are listed (these numbers can be determined by comparing the factorizations with all divisors less than 10). There are seven two-digit bombastic prime numbers, and they are marked in bold in the table.

| 11 | 13 | 17 | 19 | 23 | $\mathbf{2 9}$ | 31 | $\mathbf{3 7}$ | 41 | 43 | $\mathbf{4 7}$ | 53 | $\mathbf{5 9}$ | 61 | $\mathbf{6 7}$ | 71 | 73 | $\mathbf{7 9}$ | 83 | $\mathbf{8 9}$ | 97 |
| ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: |
| 1 | 3 | 7 | 9 | 6 | 18 | 3 | 21 | 4 | 12 | 28 | 15 | 45 | 6 | 42 | 7 | 21 | 63 | 24 | 72 | 63 |
| 1 | 3 | 7 | 9 | 6 | 29 | 3 | 37 | 4 | 26 | 47 | 35 | 59 | 6 | 67 | 7 | 37 | 79 | 38 | 89 | 79 |

In the previous list, we can notice several things related to the second part of the problem. The number 23 is not bombastic because 6 is a smaller number with the same digit product. More generally, no number containing the digits 2 and 3 can be bombastic, because by removing these two digits and adding 6 at any position, we get a smaller number with the same digit product (for example, for $\underline{2} 7 \underline{3} 7$, one of such numbers is $\underline{6} 77$).

Similarly, the number 34 is not bombastic because 26 is a smaller number with the same digit product. Therefore, no number containing the digits 3 and 4 can be bombastic (see, for example, the numbers $\underline{3} 8 \underline{4}$ and $28 \underline{6}$). Thirdly, the number 36 is not bombastic because 29 is a smaller number with the same digit product. Therefore, no number containing the digits 3 and 6 can be bombastic (see, for example, the numbers $2 \underline{3} 4 \underline{6}$ and $2 \underline{29} 4$).

On the other hand, we see that the number 38 is bombastic. The only even digit that can be with 3 in Karl's bombastic number is therefore 8.

Note. In the table in the first part of the problem, it was not necessary to consider numbers containing the digit 1 or numbers that have a larger digit in the tens place than in the units place—such numbers are never bombastic. With this insight, it was sufficient to test only eight of the listed numbers.