Math Journeys, Volume 1
A Rich, Little Problem, Part 3
We have taken an extended look into the solution and other questions related to the alphanumeric
The Solution. 2178 x 4 = 8712. We wondered if there were other multiplications that reversed a factor’s digits to form the product. One strategy we tried was to scale down and look for 3-digit examples. We found no “whole” multipliers that did so.
In 4 digits, we discovered that 1089 x 9 = 9801, and that 2178 and 8712 were also multiples of 1089. The full list of the first nine multiples reveals other reversed numbers.
1089, 2178, 3267, 4356, 5445, 6534, 7623, 8712, and 9801
By simply dividing reversals we can find the other multipliers, but they, unlike 4 and 9, are not whole. Is there a connection regardless?
6534 / 4356 = 1.5 and 7623 / 3267 = 2.33333....
We look for a pattern. The first multiple 1089 is reversed in the ninth; the second in the 8th. It helps here to know that 2.33333... is the fraction 7/3. Hmm.
One and 9 ... 2 and 8... 3 and 7.
The pattern would continue with 4 and 6 and finally 5 and 5. Ah, now we see that the multipliers are the ratios 9/1, 8/2, 7/3 and finally 6/4 gives 1.5. Our original problem and solution hid the fact that
2178 x (8/2) = 8712.
For homework you were asked to apply this relationship to 5445, the fifth multiple. Well, the terms of each ratio seem to add to ten; and 5 and 5 add to ten.
5445 x 5/5 = 5445.
Two New Directions. Yes, we can make lots more of this. Every good solution leads to at least two more good questions. Factors of 1089 suggest that we look back to smaller numbers (now that we know the multipliers are not necessarily whole), and ahead to larger ones.
1089 = 99 x 11 = 9 x 11 x 11.
Five-digit Solutions. By a combination of luck
and experience, I discovered that I could stick a 9 in the middle of the four-digit
10989 x 9 and 21978 x 4 have reversing products. Perhaps that maneuver would have been suggested if I thought of 99 elevens as just 11 short of 1100, or
99 x 11 = 1000 + 100 - 10 - 1.
What combination would make 10989?
10000 + 1000 - 10 - 1.
What factors? (10989 = 99 x 111) Is this a pattern? What about 99 x 1111 and 99 x 11,111?
Heady with my discovery, I slipped other 9s in the middle, and sure enough 109989, 1099989, 10999989, and so on, all have reversing families. It can be shown algebraically that the needed multipliers are ratios of those same numbers that add to ten, 9/1 and 8/2 and so on. Try expanding 2178 in a similar way. You will have found a second infinite set of solutions.
Smaller Solutions. If 99 x 11 and 99 x 111 generate reversing families, we should backtrack and see why we didn’t find 3-digits solutions. That’s your final assignment in this journey. Throw these patterns into reverse and look for 3-digit reversers, and find the needed ratios.
I’ll end here with a 2-digit family. It’s not really an end – just a stop while we think of more avenues to investigate or other journeys to share. I’m looking forward to them. Two-digit examples are very familiar to us and they are generated by 9 x 1. What is the reversed product and the multiplier for the factor 18? Ta ta!
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