Digits Math Question?

We can't use zero as a first digit; that would make that number really a four-digit number. So the two first digits ought to be 1 and 2 to obtain the smallest product. 

Of course, we want the smallest possible choices for the second digits, too. So we'll use 0 and 3--but which way? 

13 * 20 = 260 but

10 * 23 = 230, so lets start the numbers with 10 and 23. 

Next come the third digits; we should use 4 and 5. 

104 * 235 = 24,440 

105 * 234 = 24,570 

That makes sense; if the two numbers have the same total, then the ones with the greater DIFFERENCE have the smaller product. We can keep applying this rule to add the remaining digits: 

10,468 * 23,579 = 246,824,972 ought to be the lowest total we can get, 

assuming that using the zero as a first digit doesn't count.