Five Distinct Digits

Observe these two multiplication examples:

12 x 5 = 60

12 x 4 = 48
Do you see something uniquely different about the first case that is not true about the second one? Well, the first one is made up of five different, or distinct, digits, whereas the second one has a repeated digit, the 4.

That in and of itself may not seem so terribly significant, that is, until you try to find all the possible such cases that exist. And when this is presented to young students, say in the 3rd-5th grade levels, it becomes a reasonably decent challenge.

Initially, it can be made into a game. The rules are very simple: Find as many such cases as you can in, say 5 minutes. Score 1 point for each case found. There are more than 30 such cases possible, so there is plenty to keep the contestants busily hunting for that amount of time.

Symbolically, we are looking for solutions to multiplication statements that exhibit this mathematical structure:

AB x C = DE

Once this activity has run its course, there is always the greater challenge: go for 6 distinct digits! This means to look for multiplications that have the form of

AB x C = DEF

Happy hunting!

Update (March 2002)
Here is the complete list when all ten digits (0 – 9) are used:

There are only 22 solutions:

58401 = 63×927     32890 = 46×715     26910 = 78×345
19084 = 52×367     17820 = 36×495  &  17820 = 45×396
16038 = 27×594  &  16038 = 54×297     15678 = 39×402

65821 = 7×9403     65128 = 7×9304     34902 = 6×5817
36508 = 4×9127     28651 = 7×4093     28156 = 4×7039
27504 = 3×9168     24507 = 3×8169     21658 = 7×3094
20754 = 3×6918     20457 = 3×6819     17082 = 3×5694
15628 = 4×3907

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