remainder

juantheron · 2012-01-23 03:31:35

Find Remainder when

is divided by

Last edited by juantheron (2012-01-23 03:32:01)

John E. Franklin · 2012-01-23 03:46:53

Answer I think is 2.
See data loop:
1 Begin 2
4
8
16
32
64
27 54
7 14
28
56
11 22
44
88
75 49 98
95 89 77 53 5 10
20
40
80
59 17 34
68
35 70
39 78
55 9 18
36
72
43 86
71 41 82
63 25 50
100
99 97 93 85 69 37 74
47 94
87 73 45 90
79 57 13 26
52
3 6
12
24
48
96
91 81 61 21 42
84
67 33 66
31 62
23 46
92
83 65 29 58
15 30
60
19 38
76
51 1 CIRCLE

John E. Franklin · 2012-01-23 03:48:05

No, I think the answer is 4.

John E. Franklin · 2012-01-23 03:51:04

The loop is 101-1 or 100 long and repeats
So 0, 100, and 200, and 2**300 is 1 for the answer.
Therefore 2**101 or 2**201 has 2 for remainder.
And 2**102 and 2**202 AND 2**302 has 4 for the remainder.

John E. Franklin · 2012-01-23 03:58:35

Woops!! I think my loop should have been 101 long not 100, so I think I should have done the power residues not on the prime # 101, but on the non-prime number of 102. This might take me longer as I don't know the proper way to do non-primes, what multiplier to use...

bobbym · 2012-01-23 04:02:48

Hi;

John E. Franklin · 2012-01-23 04:04:15

I hate to be hasty, but my quick analysis of 2**y mod (3) reveals that my original answer may be correct, because it does ring around (loop around) with (3-1) or 2 numbers, so perhaps the mod(101) ring may be 100 long afterall. So I stick with my answer of 4 for the time being.

John E. Franklin · 2012-01-23 04:05:39

How did you get 2^101 mod (101) is 2, bobby?

bobbym · 2012-01-23 04:08:16

Remember Fermats little theorem? It says that if you raise a number to a prime power and take the same prime modulo you get the number itself. So 2^101 mod 101 we have 101 as prime. So the answer is 2. Then you square it to get 4.

John E. Franklin · 2012-01-23 04:10:26

Thanks for that!!! Same answer as my power residue book too!!

bobbym · 2012-01-23 04:13:03

Hi John;

Take a look at this. I know Wikipedia can be a little tough at times. It is at the top.

http://en.wikipedia.org/wiki/Fermat%27s_little_theorem

John E. Franklin · 2012-01-23 04:27:46

I am genuinely confused that they have the coprime restriction on the second equation, making it less useful. I wonder why?

bobbym · 2012-01-23 04:30:59

Hi John;

If a and p were mutiples of each other then the mods would not be one.

John E. Franklin · 2012-01-23 04:32:45

Yes but some non-co-primes are not multiples like 24 and 16, they share 2 thrice.

bobbym · 2012-01-23 04:35:10

You are right about that. Some of them would be 0 say like 32 mod 4 = 0. We would have to guess when and when not. With the coprimes it is always true.

John E. Franklin · 2012-01-23 04:37:56

I don't really complehend the whole picture of this.
What is it in summary?

bobbym · 2012-01-23 04:41:45

That is a good point. We were just talking about that in another thread. We don't really need to understand mathematics to do it. My signature was written by a great mathematician. Sometimes it is like being a great carpenter. He does not know how a hammer or a chisel is made. He knows nothing about metallurgy but he does know how to use them to make beautiful things. I usually settle for that.

John E. Franklin · 2012-01-23 04:44:51

Nice, but that doesn't help my needy brain for greater understanding. See I tend to remember things when there is logic. Maybe I should review or do somemore examples on this modular ring power thing over the next few weeks and see if I come up with any ideas... I was planning on it anyway since I have the BASIC program to fool with.

juantheron · 2012-01-23 04:46:50

Thanks to all I have also got it.

i have also use fermat Little Theorem

bobbym · 2012-01-23 04:47:07

I agree, I forget it all the time and have to refresh but until someone explains it to me in a way I can understand...

John E. Franklin · 2012-01-23 04:49:41

Hey juantheron gets it now!!!!

John E. Franklin · 2012-01-23 04:53:25

Yeah, when it "clicks", then you remember. Something doesn't even have to be proven from ground zero, but if you can prove from some easy to remember point, then you got something.

bobbym · 2012-01-23 04:54:45

I usually do not explain things unless I can put it in my own words. I hate jargon.

bobbym · 2012-01-23 04:57:04

Hi juan;

You are welcome glad to help.

John E. Franklin · 2012-01-23 04:57:32

"jargon", that reminds me it will be the year of the dragon very soon. I wonder how many days till the Chinese new year?

Math Is Fun Forum

#1 2012-01-23 03:31:35

remainder

#2 2012-01-23 03:46:53

Re: remainder

#3 2012-01-23 03:48:05

Re: remainder

#4 2012-01-23 03:51:04

Re: remainder

#5 2012-01-23 03:58:35

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#6 2012-01-23 04:02:48

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#8 2012-01-23 04:05:39

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#9 2012-01-23 04:08:16

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#10 2012-01-23 04:10:26

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#11 2012-01-23 04:13:03

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#12 2012-01-23 04:27:46

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#13 2012-01-23 04:30:59

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#14 2012-01-23 04:32:45

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#16 2012-01-23 04:37:56

Re: remainder

#17 2012-01-23 04:41:45

Re: remainder

#18 2012-01-23 04:44:51

Re: remainder

#19 2012-01-23 04:46:50

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#21 2012-01-23 04:49:41

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#22 2012-01-23 04:53:25

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#23 2012-01-23 04:54:45

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#24 2012-01-23 04:57:04

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