Need Help in Complete Division of 987654321 / 123456789

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param-97

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Hello all Experts,
This is my first Post on this Forum.
I am a Math learner. Well, I am not a mathematician but I want to get some information in the field of mathematics. The main reason for me to here is that I am completely stopped at a point in recreational math research. I need to find the repeatend of 987654321 / 123456789. I searched a lot on the deep internet for its answer but I could not find any information. Few days ago, I Installed Microsoft Excel Add-In but This division answer seems to be a few more digits than normal decimal expansion in my capacity. Can anyone help me with this matter ? On big calculators like comptune.com/calc.php It didn't stops even at full capacity of 1 Million digits of decimal expansion. My concern is where it's actual repeatend is located ? I am providing some digits of 987654321 / 123456789 division.

987654321 / 123456789 =
8.00000007290000066339000603684905493532639991147023919437917666885050768653961994751054......

Where its actual repeatend located where this decimal expansion repeats?
 
param-97 said:
I need to find the repeatend of 987654321 / 123456789. I searched a lot on the deep internet for its answer but I could not find any information. Few days ago, I Installed Microsoft Excel Add-In but This division answer seems to be a few more digits than normal decimal expansion in my capacity. Can anyone help me with this matter ? On big calculators like comptune.com/calc.php It didn't stops even at full capacity of 1 Million digits of decimal expansion. My concern is where it's actual repeatend is located ? I am providing some digits of 987654321 / 123456789 division.

987654321 / 123456789 =
8.00000007290000066339000603684905493532639991147023919437917666885050768653961994751054......

Where its actual repeatend located where this decimal expansion repeats?
Click to expand...

What is a "repeatend"? Are you using this term to refer to the line of repeated digits that any rational division eventually coughs up?
 
param-97 said:
Hello all Experts,
This is my first Post on this Forum.
I am a Math learner. Well, I am not a mathematician but I want to get some information in the field of mathematics. The main reason for me to here is that I am completely stopped at a point in recreational math research. I need to find the repeatend of 987654321 / 123456789. I searched a lot on the deep internet for its answer but I could not find any information. Few days ago, I Installed Microsoft Excel Add-In but This division answer seems to be a few more digits than normal decimal expansion in my capacity. Can anyone help me with this matter ? On big calculators like comptune.com/calc.php It didn't stops even at full capacity of 1 Million digits of decimal expansion. My concern is where it's actual repeatend is located ? I am providing some digits of 987654321 / 123456789 division.

987654321 / 123456789 =
8.00000007290000066339000603684905493532639991147023919437917666885050768653961994751054......

Where its actual repeatend located where this decimal expansion repeats?
Click to expand...
I suppose you found things like this:

mathlesstraveled.com

Finding prefix and repetend length

We interrupt your regularly scheduled primality testing to bring you something else fun I’ve been thinking about. It’s well-known that any rational number $latex a/b$ has a decimal expansion that e…
mathlesstraveled.com mathlesstraveled.com
mathlesstraveled.com

Finding the prefix length of a decimal expansion

Remember from my previous post that we’re trying to find the prefix length $latex k$ and repetend length $latex n$ of the decimal expansion of a fraction $latex a/b$, that is, the length of the par…
mathlesstraveled.com mathlesstraveled.com
mathlesstraveled.com

Finding the repetend length of a decimal expansion

We’re still trying to find the prefix length $latex k$ and repetend length $latex n$ of the decimal expansion of a fraction $latex a/b$, that is, the length of the part before it starts repeating, …
mathlesstraveled.com mathlesstraveled.com
 
987654321 / 123456789 = 8 + 9/123456789 = 8 + 1 / (3607*3803), and since 3607 and 3803 are both prime we know that the period would divide gcd (3607-1, 3803-1) = 6855006. It took my script 2.5 hours to verify that there is no smaller period than that. But I'll leave it to you to figure out where the repeated sequence actually starts.
 
987654321 / 123456789 = 8 + 9/123456789 = 8 + 1 / (3607*3803), and since 3607 and 3803 are both prime we know that the period would divide gcd (3607-1, 3803-1) = 6855006.
Click to expand...
Sir,
It is a very clear answer. But I want to know a very simple question. Is there is clear cut way to find just the repeated sequence number (repetend) *not the decimal expansion digits physical in front of eyes* by applying long division method in Excel or in any language in other softwares. A Yes or No will be very helpful for me if any one knows or doing the same with their handy tools.
 
Last edited:
param-97 said:
Sir,
It is a very clear answer. But I want to know a very simple question. Is there is clear cut way to find just the repeated sequence number (repetend) *not the decimal expansion digits physical in front of eyes* by applying long division method in Excel or in any language in other softwares. A Yes or No will be very helpful for me if any one knows or doing the same with their handy tools.
Click to expand...
Sorry, but I am not familiar with Excel or other tools except GNU GMP library and its Python interface gmpy2, which I used.
 
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