+633 I was reading this book and kept noticing that one number kept popping up - 1089. It turns out that when you multiply it, the numbers move sequentially through the columns.
1 - 1089
2 - 2178
3 - 3267
4 - 4365
5 - 5454
6 - 6543
7 - 7632
8 - 8721
9 - 9810
Can anyone else think of some?
Yes !! 142857
142857 x 1 =142,857
142857 x 2 =285,714 same numbers in different order.
142857 x 3 =428,571 ...................................................
142857 x 4 =571,428.....................................................
142857 x 5 =714,285......................................................
142857 x 6 =857,142......................................................
142857 x 7 =999,999 !!!
+2446 Wow! This pattern that you have displayed actually continues past 7, actually, as I discovered just now. You just have to perform an intermediary manipulation.
142857∗8=1142856⇒1+142856=142857
142857∗9=1285713⇒1+285713=285714
142857∗10=1428570⇒1+428570=428571
It seems to be that any multiple of 7 results in the 999999 result while anything else results in this "cyclic" property.
142857∗77=10999989⇒10+999989=999999
Here are a few arbitrary attempts I tried to see if the pattern maintained itself. To my relief, it does.
142857∗43=6142851⇒6+142851=142857
142857∗6807=972427599⇒972+427599=428571
142857∗142857=20408122449⇒20408+122449=142857
Amazingly, this pattern maintains itself for incredibly large multipliers. It requires more manipulation, though. This is incredible!
142857∗758241142857=108320054945122449⇒108320054945+122449=108320177394108320177394⇒108320+177394=285714
+633 Thought of another one: 76923
Multiples:
10769231076923096923071292307632307694307692&2153846753846153846151184615364615388615384
+2446 helperid1839321, I decided to delve deeper into this subject. You may like my findings!
First of all, I think I understand why this property occurs. I happen to know that 142857 is the first 6 digits of the decimal expansion of 17. At first, I thought this was insignificant, but it turns out that this fact can be used to understand this further. Look at the table below.
| 17=¯0.142857142857 | Multiply both sides by 10. |
| 107=1.42857¯142857 | Let me rewrite 10/7 to make things clearer. |
| 1+37=1.42857¯142857 | Subtract one from both sides. |
| 37=0.42857¯142857 | WOAH! It cycles! Let's do this again. Multiply both sides by 10 again. |
| 307=4.2857¯142857 | Rewrite 30/7 again. |
| 4+27=4.2857¯142857 | Subtract 4 from both sides. |
| 27=0.2857¯142857 | WOAH! The first 6 digits cycle again. You can continue the pattern, if you wish! |
This forced me to wonder if there is any more solutions for 1p, where p is a whole number that creates this special property. This is because if another number p causes some repetition, then we would have found another number! YAY!
I decided to enlist some help from a computer here. This is what the computer outputted.
7, 17, 19, 23, 29, 47
WHAT! There are more! Yes, these have the same property. Let's check them out, shall we?
| p | 1p | ||
| 7 | .142857... | ||
| 17 | .0588235294117647... | ||
| 19 | .052631578947368421... | ||
| 23 | .0434782608695652173913... | ||
| 29 | .0344827586206896551724137931... | ||
| 47 | .0212765957446808510638297872340425531914893617... | ||
It appears as if 142857 is the only number that does not start with a zero. Let's keep running this simulation! Thankfully, more numbers output! I let it run for some time, too.
59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193...
In case you are wondering,
1193=005181347150259067357512953367875647668393782383419689119170984455958549222797927461139896373056994818652849740932642487046632124352331606217616580310880829015544041450777202072538860103626943...
There are a few patterns that I see here
1) p must be a prime number
2) The decimal expansion must have a maximum period decimal expansion of p−1.
I am also making a conjecture here that I do not know whether or not is true: there are an infinite number for p that create these types of numbers!
