 # FIVE DISTINCT DIGITS Observe these two multiplication examples:

 12x 560 12x 448
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:

 ABx CDE

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

 ABx CDEF

Happy hunting!

### Update (March 2002)

Here is the complete list when all ten digits (0 - 9) are used:

There are only 22 solutions:

```58401 = 63x927     32890 = 46x715     26910 = 78x345
19084 = 52x367     17820 = 36x495  &  17820 = 45x396
16038 = 27x594  &  16038 = 54x297     15678 = 39x402

65821 = 7x9403     65128 = 7x9304     34902 = 6x5817
36508 = 4x9127     28651 = 7x4093     28156 = 4x7039
27504 = 3x9168     24507 = 3x8169     21658 = 7x3094
20754 = 3x6918     20457 = 3x6819     17082 = 3x5694
15628 = 4x3907
```