what numbers are most likely to be the longest cycle reciprocal?

Question

example of what I mean 1/7 has a 6 remainders
.142857 all repeat.

This is true of all xx/7
22/7=3+
14857 all repeat

43/7=4+
1482857 all repeat

Answer 1

This is actually a good question! Suppose 1/p has a cycle length of n. Then (1/p)*10^n -(1/p) will be an integer, so p divides 10^n -1. It is pretty clear that the converse holds also; that is, if p divides 10^n -1, then 1/p cycles after a stretch of length n. So what we really want is the smallest n so that p divides 10^n -1. This is 'the order of 10 in the group of units modulo p'.

For example, since 11 divides 10^2 -1=99 and not 10^1 -1=9, we see that 1/11 repeats after 2 steps. Now, it is known that if k is the number of integers less than p that have no factors in common with p, then 10^k -1 is divisible by p. It is also known that the smallest n above will divide this value of k. So for example, with p=11, k=10, but n=2. Notice that 2 does divide 10.

Because of the nature of k, it is good to look at values of p that are prime. For example, p=59 has a repeat length of 58. It is easy to see that 1/17 has a cycle length of 16.

Answer 2

That's because it is a 'rational' number. One that can be expressed as the quotient of two integers.

But the 'irrational' numbers (such as π and √2) have decimal expansions that never repeat and never end.

Why do
22/7 = 3.1428571428571428571428571428571
and
43/7 = 6.1428571428571428571428571428571
have the same decimal parts?

It's because 43-22 = 21 which is an exact multiple of 7. In fact 21 = 7*3 and the difference in the integer parts is exactly 3.

But it's good that you noticed that. Why? Because in higher math the remainder is sometimes *much* more important then the quotient. Especially in things like number theory.


Doug

Answer 3

What about 22/7?

Answer 4

In general, if the number does not have 2 or 5 as a factor, then the larger the number is, the longer the sequence *could* be. I don't know if it always is longer though.

For a number n, the sequence will never be longer than n. Or mayb it's n-1. When you get old like me, a lot of information starts falling out of your memory :(

Answer 5

1/23= .043478208695652173613 repeating
21#'s repeating

1/31= .032258064516129032258064516129 repeating
30#'s

Answer 6

a prime number

find the largest prime number and the make a reciprocal of it.

From what I have done the larger the number
longer the cycle is if it is a prime number

may have a answer to question

http://www.answers.com/main/ntq-tname-unique%252Dprime-fts_start-

Answer 7

68/7=9.7 and repeating 14286