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Fibonacci Mystery - Numberphile - YouTube
Channel: Numberphile
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today I want to do a video response I
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response to one of our own numberphile
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videos because some time ago Brady made
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a video with our numberphile composer
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Alan Stuart's it was a video it was 40
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minutes long this video which was a test
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to our loyalty I think but in that video
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have you made it through alan described
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composing one of his pieces about the
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Fibonacci sequence it involved the
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Fibonacci sequence and he came up with a
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question as a challenge I guess to the
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number files I'm not quite good enough
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in laughs to be able to explain why that
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is maybe a number file will be able to
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explain like this I'm going to answer
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his question today
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so what Alan was doing was he was trying
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to involve the Fibonacci sequence in a
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piece of music so first let's have a
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recap of the Fibonacci sequence very
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important sequence in mathematics it's
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quite easy it starts with 1 then it's 1
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again and then you add the previous 2
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value so you add these two and you get
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the next number so that's just 1 plus 1
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equals 2 then you add these two values
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that's 1 plus 2 equals 3 then you add
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the previous 2 that's 2 plus 3 equals 5
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and you keep going in this way 6760 5
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and so on okay so there's the Fibonacci
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sequence and what Allan was doing was to
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turn this into a piece of music he was
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dividing by 7 he's going to divide all
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these Fibonacci numbers by 7
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now you can't divide exactly by 7 most
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of the time and so he looked at the
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remainder what you get left over when
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you divide by 7 that's one that's two
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that's three that's 5 now 8 when you
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divide 8 by 7 that's one with one left
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over that's what he wrote down then 13
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divided by 7 is 1 with 6 left over 21
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divided by 7 is 3 exactly it has zero
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left over so you write zero and we're
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going to continue in that way we're
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going to do all the remainders so he
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wrote out all the remainders I need to
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make it a piece of music he turned to
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these remainders into musical notes so
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it corresponds to notes but he didn't
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notice a pattern when he did it
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curiously he found the pattern repeated
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it actually repeated every 16 numbers so
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here we go 1 1 2 3 5 1 6 0 6 6 5 4 - 6 1
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0 1 1 2 3 and then the pattern repeated
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and he asked what is this about is this
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a thing and it is a thing it's actually
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called a Poisson Oh Pierre
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pisano period is named after Leonardo
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Pisano and if you think you haven't
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heard of him before are you wrong
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because that's just another name of
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Fibonacci so this is a pisano period now
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Allen chose seven to do this because
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that helps him make his musical notes if
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you'd pick any number though he would
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have got up here as well but the length
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of the period would have been different
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so he divided by 2/3 divided by 90 or
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700 he would have got a cyclic pattern
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if we divide by 5 as an example this
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should be quite easy divided by 5 this
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is when I'll make a mistake 1 1 2 left
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over 3 divided by 3 5 plus 3 left over 5
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divided by 5 is 0 0 left over 8 divided
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by 5 is a 1 a 3 left over would be a 3 4
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left over that's one left over a 0 left
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over and I think that's actually the
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full pattern and the when you divide by
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5 the period is 20 so this is the whole
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thing here at this point it will start
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to repeat again the zeros I want to
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point out are interesting the zeros are
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when you can divide by 5 exactly so
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there's a 0 here there's a 0 here this
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is 0 here as well oh look at this no
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look I just noticed that laser mistake I
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said that look I said this had a
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remainder of 5 but look I don't need
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this that has a remainder of 0 that just
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shows you doing it all alive I I'm doing
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it up now I noticed that that was a
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mistake I noticed there was a problem
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there because I knew that the period has
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only can only have one zero two zeros or
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four so that's another result in
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mathematics so because there was three I
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knew there was a problem there and I
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found it so that's another example of
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the Pisano
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period if you another thing I want to
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point out then it's actually every fifth
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number every fitnah but there is a there
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is a proper bit of
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it's behind this room actually do it
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there's another result about Fibonacci
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numbers it says a Fibonacci number let's
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call it the nth Fibonacci number
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exactly divided which I use a vertical
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line for another Fibonacci number which
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I'm going to call FM the result is if
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and only if n divides M so the little
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index at the bottom here if they divide
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then the Fibonacci numbers divide as
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well just to give you an example then we
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were doing if we look at the fifth
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Fibonacci number that is equal to 5 so
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so 5 divides Fibonacci numbers if and
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only if there's a little symbol there 5
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divides the value of the index so every
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fifth number every 5th Fibonacci number
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is divisible by 5 that's what I chose
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let me just do one more example of that
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to make the point I took the next one
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look the 6th Fibonacci number is 8 so 8
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will divide Fibonacci numbers exactly
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when 6 that's the index here divides M
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so every sixth number every sixth
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Fibonacci number is divisible by 8 so
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this Pisano period idea was first
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discovered by a mathematician called
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Lagrange in 1774 and he was actually
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looking at the patterns when you divide
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by 10 when he divide by 10 what you have
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left over it's just the last digit let's
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do a quick version of that what colors
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do I need now I have to use black 5 8 13
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divided by 10 is 1 with 3 left over 21
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divided by 10 has 1 left over
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and what he discovered was the they had
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8 they did have a pattern and the
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pattern had a period of 60 so the last
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digits of the Fibonacci sequence have a
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pattern of length 60 if you divide by a
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hundred then you're actually looking at
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the last two digits of the Fibonacci
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sequence that has a pattern of 300
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lengths 300 if you look at the last
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three digits that's dividing by a
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thousand
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that has a pattern of length 1,500 the
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other easy way to do this is you don't
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have to write out the Fibonacci sequence
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and do each calculation for each number
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actually there's a property of these
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remainders that you can just add up the
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previous two remainders let's have a
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look so if we look at an example like
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here this is when I was dividing by 7
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for the remainder of 4 plus a remainder
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of 2 gives me the next remainder it's
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just like the Fibonacci sequence gives
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me a remainder of 6 remainder of 2 plus
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a remainder of 6 it actually wraps back
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because we're dividing by 7 when you go
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past 7 it wraps back to 0 so this will
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take me back to 1 then 6 plus 1 will
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give me 7 which wraps back to 0 so alack
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I can actually do these sequences by
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simply adding up the remainders so using
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this idea when you add the previous two
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remainders gives you the next one
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actually kind of explains what's going
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on when I had a 1 and a 0 here next to
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each other then when I add them together
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I get a 1 then I add 0 plus 1 and I get
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another one so I've got 2 ones in a row
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1 and 1 and then that's the beginning of
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the Fibonacci sequence and then it just
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carries on so whenever you have a 0 next
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to a 1 you're going to end up back into
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the Fibonacci so you're going to end up
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back to the start of the Fibonacci
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sequence and it has been shown that this
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will always happen so that 0 1 is like a
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big trigger point yeah yeah that's going
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to send you back to the start again and
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that will happen in the other sequences
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that we had here as well look this is
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the dividing by 5 it has a period of 20
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that's the end of it and there's the 0
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and the 1 again and that's going to send
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you back to the start so you're always
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going to end up back to the start this
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period is always going to repeat however
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there is no general formula for the
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length of the period so that's something
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we don't know yet
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that two boxes of nine and a box of six
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there we go that makes 44
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