The decimal expansion of 1/271 repeats with a period of length 5:
.003690036900369 ...
However, it is not the smallest number q for which the decimal expansion of 1/q has a repetition length of 5.
Find the smallest q so that the decimal expansion of 1/q has repetition length n for each of {1, 2, ..., 10}
Is there a simple way of finding such a number?
(In reply to
computer exploration by Charlie)
By doing the division into 1 by the long division algorithm instead of by the internal workings of the computer, the table can be extended:
rep length prime fact of rep-9 smallest q factors of smallest q
2 3 3 11 11 11
3 3 3 3 37 27 3 3 3
4 3 3 11 101 101 101
5 3 3 41 271 41 41
6 3 3 3 7 11 13 37 7 7
7 3 3 239 4649 239 239
8 3 3 11 73 101 137 73 73
9 3 3 3 3 37 333667 81 3 3 3 3
10 3 3 11 41 271 9091 451 11 41
11 3 3 21649 513239 21649 21649
12 3 3 3 7 11 13 37 101 9901 707 7 101
13 3 3 53 79 265371653 53 53
14 3 3 11 239 4649 909091 2629 11 239
15 3 3 3 31 37 41 271 2906161 31 31
It's still undetermined how the choices of prime factors are made, such as 7 * 101 = 707 for length 12.
The long division code:
q$ = "": dnd$ = "1"
DO
frst = 1
WHILE VAL(dnd$) < prod
dnd$ = dnd$ + "0"
IF frst = 0 THEN q$ = q$ + "0"
frst = 0
WEND
q = INT(VAL(dnd$) / prod)
r = VAL(dnd$) - q * prod
dnd$ = STR$(r)
q$ = q$ + LTRIM$(STR$(q))
LOOP UNTIL LEN(q$) > 25
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Posted by Charlie
on 2006-09-25 11:35:02 |
re: computer exploration
