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
i would what to find out ITXIT=FIT STAND FOR IN THE DISTINTIVE DIGIT?