# Find the repetend

• May 15th 2010, 03:22 AM
Mariner
Find the repetend
a) find the repetend for the fraction 1/103

b) find the five largest fractions amongst n/103 for n = 1,2,3.....,102 whose repetends have the cyclic order as the repetend for 1/103.

c)find three fractions amongst n/103 for n = 1,2,3,....,102 whose repetends do not have the same cyclic order.

d) represent the three different repetends in part c as A,B,C. Determine which of these repetends each of the fractions n/103 has for n = 1,2,3,....,10 and 93,94,95...,102. what pattern do you notice?
• May 15th 2010, 05:38 AM
TKHunny
Why is this difficult? You just have to stick with it until you find the first repeated remainder.

I get this: 0097087378640776699029126213592233

You may wish to use a spreadsheet or write a small program to do this.

Now what?
• May 15th 2010, 02:07 PM
Mariner
Quote:

Originally Posted by TKHunny
Why is this difficult? You just have to stick with it until you find the first repeated remainder.

I get this: 0097087378640776699029126213592233

You may wish to use a spreadsheet or write a small program to do this.

Now what?

I dont understand(Worried) I know for the first part I have to rewrite it as a decimal. but what about all the other steps?
• May 15th 2010, 09:23 PM
NowIsForever
This is what I found. I Googled bignum calculator, obtained software and constructed this table. You could do something similar.

102 9902912621359223300970873786407766 1
101 9805825242718446601941747572815533 2
100 9708737864077669902912621359223300 1
99 9611650485436893203883495145631067 4
98 9514563106796116504854368932038834 4
97 9417475728155339805825242718446601 2
96 9320388349514563106796116504854368 4
95 9223300970873786407766990291262135 1
94 9126213592233009708737864077669902 1
93 9029126213592233009708737864077669 1
92 8932038834951456310679611650485436 4
91 8834951456310679611650485436893203 4
90 8737864077669902912621359223300970 1

1 0097087378640776699029126213592233 102
2 0194174757281553398058252427184466 101
3 0291262135922330097087378640776699 102
4 0388349514563106796116504854368932 99
5 0485436893203883495145631067961165 99
6 0582524271844660194174757281553398 101
7 0679611650485436893203883495145631 99
8 0776699029126213592233009708737864 102
9 0873786407766990291262135922330097 102
10 0970873786407766990291262135922330 102

102, 100, 95, 94, 93, 90, ..., 10, 9, 8, 3, 1
101, 97, ..., 6, 2
99, 98, 96, 92, 91, ..., 7, 5, 4
• May 16th 2010, 12:00 AM
Mariner
Quote:

Originally Posted by NowIsForever
This is what I found. I Googled bignum calculator, obtained software and constructed this table. You could do something similar.

102 9902912621359223300970873786407766 1
101 9805825242718446601941747572815533 2
100 9708737864077669902912621359223300 1
99 9611650485436893203883495145631067 4
98 9514563106796116504854368932038834 4
97 9417475728155339805825242718446601 2
96 9320388349514563106796116504854368 4
95 9223300970873786407766990291262135 1
94 9126213592233009708737864077669902 1
93 9029126213592233009708737864077669 1
92 8932038834951456310679611650485436 4
91 8834951456310679611650485436893203 4
90 8737864077669902912621359223300970 1

1 0097087378640776699029126213592233 102
2 0194174757281553398058252427184466 101
3 0291262135922330097087378640776699 102
4 0388349514563106796116504854368932 99
5 0485436893203883495145631067961165 99
6 0582524271844660194174757281553398 101
7 0679611650485436893203883495145631 99
8 0776699029126213592233009708737864 102
9 0873786407766990291262135922330097 102
10 0970873786407766990291262135922330 102

102, 100, 95, 94, 93, 90, ..., 10, 9, 8, 3, 1
101, 97, ..., 6, 2
99, 98, 96, 92, 91, ..., 7, 5, 4

Is that how you do the question?
• May 16th 2010, 08:00 AM
NowIsForever
Quote:

Originally Posted by Mariner
Is that how you do the question?

Of course, there's more to it. Consider the question from your earlier post: "what pattern do you notice?"

Furthermore, you can extend the table and look for patterns using numbers other than 103.

Each of the above cycles has 34 digits. Do you think that is somehow related to the number 103? If so, how? Is it significant that 103 is a prime number?