## 1. How to compute a repeating sequence

### A notation of repeating sequence and computation method

In this chapter we compute the repeating sequence and its length for circulating fraction of 1/p where p is prime up to 100 except 2 and 5.
For example,

1/7 = 0.[142857], length=6
1/11 = 0.[09], length=2
1/13 = 0.[76923], length=6
...

Formal notation of repeating decimal is to put a dot on the begin and end of repeating letter. But it is inconvenient for HTML notation, so in this chapter we donote as above, that is, locating repeating decimals between  [ ] .

### Program

When we compute 1 divede by 7 by hand, we do as follows;

1/7=0 ... 1
put "0" at the right of "1", compute 10/7
10/7=1 ... 3
put "0" at the right of "3", compute 30/7
......

repeating the procedure that "put 0 at the right of previous remainder and divide by 7".
When the remainder is equal to "1", the next circulation starts.
The remainders dividing by p are,

1, 2, ... , p-1

so the length of repeating decimals is at most p-1.

```10   ' cycle
20   A=1:P=7
30   B=A*10\P:A=res:print B;:if A<>1 then 30
```

We compute the repeating sequence and its length for
circulating fraction of 1/p where p is prime up to 100 except 2 and 5.

``` 5   ' cycle_p
10   for I=1 to 25:N=prm(I):if or{N=2,N=5} then 100
20     int "n=";N
30     A=1:Length=0
40     B=A*10\N:A=res:print B;:inc Length:if A<>1 then 40
50     print:print Length
60   next I
```

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E-mail : kc2h-msm@asahi-net.or.jp
Hisanori Mishima