5623109 = 23 x 41 x 67 x 89

There are in total 12 ways of combining four two-digit primes with no digits in common, but only one of the twelve products contains no repeating digits.

This is how you can reduce the number of possible arrangements to just twelve (which can then be easily checked to see if the product contains no repeated digits), just with a little logic:

Once we are past single digit primes, all primes must end in 1, 3, 7, or 9. Since we are after four primes with no repeated digits, they must end in these four digits, and therefore they cannot start with those four digits. They also cannot start with a 0 as they wouldn’t be two-digit numbers, so the start numbers must be four of the following five: 2,4,5,6,8. Since the primes starting with 2,5,8 all end in 3 or 9, only two of those three start numbers will be used, and the 3 and 9 will also come from those. Therefore 4 and 6 must be in any selection, and must be followed by 1 and 7.

In summary, two of the numbers must start with 2,5,8 and end with 3,9 (6 possibilities), and two must start with 4,6 and end with 1,7 (2 possibilities). Overall there are therefore 6x2= 12 possibilities.