+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.
\(142857*8=1142856\Rightarrow1+142856=142857\)
\(142857*9=1285713\Rightarrow1+285713=285714\)
\(142857*10=1428570\Rightarrow1+428570=428571\)
It seems to be that any multiple of 7 results in the 999999 result while anything else results in this "cyclic" property.
\(142857*77=10999989\Rightarrow10+999989=999999\)
Here are a few arbitrary attempts I tried to see if the pattern maintained itself. To my relief, it does.
\(142857*43=6142851\Rightarrow6+142851=142857\)
\(142857*6807=972427599\Rightarrow972+427599=428571\)
\(142857*142857=20408122449\Rightarrow20408+122449=142857\)
Amazingly, this pattern maintains itself for incredibly large multipliers. It requires more manipulation, though. This is incredible!
\(142857*758241142857=108320054945122449\Rightarrow108320054945+122449=108320177394\\ 108320177394\Rightarrow108320+177394=285714 \)
+633 Thought of another one: 76923
Multiples:
\(\begin{matrix} 1 && 076923 \\ 10 && 769230 \\ 9 && 692307 \\ 12 && 923076 \\ 3 && 230769 \\ 4 && 307692 \end{matrix} \: \& \: \begin{matrix} 2 && 153846 \\ 7 && 538461 \\ 5 && 384615 \\ 11 && 846153 \\ 6 && 461538 \\ 8 && 615384 \end{matrix}\)
+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 \(\frac{1}{7}\). 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.
| \(\frac{1}{7}=\overline{0.142857142857}\) | Multiply both sides by 10. |
| \(\frac{10}{7}=1.42857\overline{142857}\) | Let me rewrite 10/7 to make things clearer. |
| \(1+\frac{3}{7}=1.42857\overline{142857}\) | Subtract one from both sides. |
| \(\frac{3}{7}=0.42857\overline{142857}\) | WOAH! It cycles! Let's do this again. Multiply both sides by 10 again. |
| \(\frac{30}{7}=4.2857\overline{142857}\) | Rewrite 30/7 again. |
| \(4+\frac{2}{7}=4.2857\overline{142857}\) | Subtract 4 from both sides. |
| \(\frac{2}{7}=0.2857\overline{142857}\) | 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 \(\frac{1}{p}\), 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?
| \(p\) | \(\frac{1}{p}\) | ||
| 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,
\(\frac{1}{193}\)=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!
