The real last digit is 6 or 8. So the real last digit of "466" can't be 5, as no multiple of 5 ends in [6,8]. So "466" is "7". But that makes "5125" "4" because if it ended in 6 the final digit of the product would be 2.
Of the four possibilities for the last pairs of digits in the factors, we have "14", "34" x "57", "77", and the tens digit of the product must be in [7,9]. Well, that rules out "34" because 34x57 and 34x77 both have forbidden tens digits. And so "5125" is "14". But so far both options for the tens digit of "466" are fair game.
Consider the 100s digits. we have [0,2]x[3,5] + (carry from tens) = [6,8]. Now the tens situation is either 14x57 (=798 so carry of 7) or 14x77 (=1098, carry of 10). So we have [0,2]x[3,5] + [7,10] ends in [1,3]. Clearly 0 isn't an option for the first choice, so "5125" = "214", and 2*[3,5] + [7,10] ends in [1,3]. Again, 5 is impossible for the 2nd choice, so "466" = "37" and we have 2*3 + [7,10] ends in [1,3]. Well, that only works if we pick 7, since 6+7 ends in 3, and so the 2nd factor's tens digit is 5 (since 14*57 = 798). And so we have "214" * "357" = "398".
Well now there's just two options for the first digit of the first factor, and trial and error is enough to see that only "6" works. So the factors are 6214 and 357 and their product is 2218398.

Posted by Paul
on 20140422 18:46:22 