Why does division of any number by 43 lead to an unusual but similar repeating decimal pattern?

Question

Make sure your calculator shows more than 20 digits. After about 10 or so digits the patterns repeats itself. This excludes 0 and 1. Does this have to do with the fact that 43 is a prime number? I tried many combinations but saw a pattern in every case.

And it excludes 43.

Answer 1

Every decimal number that is a quotient of two rational numbers must eventually repeat or terminate. Otherwise it would be irrational.

Check out the discussion on Wikipedia:

http://en.wikipedia.org/wiki/Recurring_decimal

Answer 2

Any decimal expansion of a fraction in base 10 by a number which has primes other than 2 and 5 will have repeating sequences. For example, 1/666 = 0.00150150150150150150150150....

Division by 7 is interesting because the digits that make up the repeating sequence is the same, regardless of what it's divided into.

1/43 = 0.023255813953488372093023255813953488372093...., so I'm not sure what's unique about this.

Answer 3

Case 1: n is a multiple of 43. n/43 divides equally with no fractional part.

Case 2: n is not a multiple of 43. Then the remainder when n is divided by 43 is an integer between 1 and 42. Since all of those values are relatively prime to 43, when you divide it out, you get a repeating decimal.

The same thing will work for any prime that is not 2 or 5.

Answer 4

I din't use a calculator. I suspected that there's some sort of repeatition that shows up in long division, I was correct.

Do the long division problem

10000/43

100-2*43=44
44-43=1

See what will happen? When you drop down 2 more zeros to carry on the division, you get the same repeatition.

Try it with another starting numerator and you'll see that you get a repeating sequence of subtractions....

So, what's involved here is an interaction between our base 10 number system and the number. Let p=43

100-2*p = p+1

Answer 5

Rational numbers give repeating decimal fractions unless the division is by a number whose prime factors are 2 and 5 (since we use base 10).

It's not as interesting as 7's. Look at the first 6 and next 6 digits.

You should also get interesting results fro 21's (the smallest number with two different prime factors that are not 2 or 5).

Answer 6

It's not unique to 43. It can be any number that has a factor other than 2 or 5. There will always be a pattern
ex:
1/7=0.14285714285714285714285714285714 (142857 repeats
1/21=0.047619047619047619047619047619048 (476190 repeats)

Answer 7

Be sure to exclude all multiples of 43 as well.