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 Repeating decimals (Posted on 2006-09-25)
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?

 See The Solution Submitted by Jer Rating: 4.5000 (2 votes)

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 re: computer exploration | Comment 2 of 11 |
(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

 Posted by Charlie on 2006-09-25 11:35:02

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