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Wonders of math

by mahesh_2961
Tags: math, wonders
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mahesh_2961
#1
Dec19-04, 05:16 AM
P: 20
hi ,
i recently found while playing around with my new calculator that any number repeated six times is divisible by 7 ....
for eg, 4 repeated 6 times gives 444444/7 =63492
an say 761 repeated 6 times 761761761761761761/ 7 =108823108823108823
the last one was done using a computer ...
can anyone plz help me out why this happens???

regrds
Mahesh
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mahesh_2961
#2
Dec19-04, 05:17 AM
P: 20
one correction to my query ... not any number but any integer
sorry for that

Mahesh
Zurtex
#3
Dec19-04, 06:23 AM
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22715371505222715371505222715371505222715371505222715371505222715371505 2 is 227153715052 repeated 6 times and is not a multiple of 7 as its prime factors are:

2 ^ 2 x 3 x 11 x 19 x 37 x 67 x 73 x 137 x 3169 x 52579 x 98641 x
333667 x 2082527 x 99990001 x 999999000001 x 3199044596370769

Edit: Sorry got a little carried away on my computer

mahesh_2961
#4
Dec19-04, 06:37 AM
P: 20
Wonders of math

hi ,
repeating an integer, x (say), six times is not the same as multiplying it six times ....
i m sorry but i think u havent got my question correctly...
it is like this
suppose x is an n digit number
repeating it 6 times gives a number
y= 1*x + 10^(n)*x + 10^(2n)*x + 10^(3n)*x+ ....+10^(6n)*x
which is not equal to x^6
and i think i have got the solution for this problem....
if u want me to post it plz tell me , or if u want to find it urself then its good for u

regards
Mahesh
gonzo
#5
Dec19-04, 06:41 AM
P: 277
well, it's easy to prove for any one digit number n:

10^5*n+10^4*n+10^3*n+10^2*n+10*n+n

That is a single digit n repeated 6 times, divide by 7 and you get:

14285 5/7 * n +
1428 4/7 * n +
142 6/7 * n +
14 2/7 * n +
1 3/7 * n +
1/7 * n =

15870*n + (21/7)*n = 15873 * n

So, a single digit repeated 6 times will always divide by 7 to an integer. In other words, if you multiply 15873 * 7 * n , where n is a single digit, then you will get a number that is that digit repeated 6 times.

If you feel like it, you can attempt similar proofs for more than one digit to see if it holds for more than one digit repeated, and if not than maybe you can find certain conditions for which it would hold with bigger numbers.

Or maybe someone has a better way to prove this than me ...
Muzza
#6
Dec19-04, 06:59 AM
P: 696
It does not appear to be true for 123423123423123423123423123423123423 either (that is 123423 repeated 6 times).
matt grime
#7
Dec19-04, 07:31 AM
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Repeating a string, s, of r digits six times is the sam as the sum of a geometric progression. Let k be 10^r, it is

s +ks +k^2s+..+k^5s

The sum of this is (s/9)*(10^6-1)

since 9=2 mod 7, and thus invertible, and 10 is coprime with 7, this reduces to zero mod 7 quite trivially by fermat's little theorem.
gonzo
#8
Dec19-04, 07:32 AM
P: 277
Quote Quote by mahesh_2961
hi ,
repeating an integer, x (say), six times is not the same as multiplying it six times ....
i m sorry but i think u havent got my question correctly...
it is like this
suppose x is an n digit number
repeating it 6 times gives a number
y= 1*x + 10^(n)*x + 10^(2n)*x + 10^(3n)*x+ ....+10^(6n)*x
which is not equal to x^6
and i think i have got the solution for this problem....
if u want me to post it plz tell me , or if u want to find it urself then its good for u

regards
Mahesh
You do realize this post doesn't make much sense, do you? First, the person you responded to didn't multiple a number 6 times, he/she did what you did, and just repeated one.

And your formula is wrong. It should only go up to 10^(5n).
gonzo
#9
Dec19-04, 07:39 AM
P: 277
Quote Quote by matt grime

since 9=2 mod 7, and thus invertible
Matt, could you explain this snippet a bit more for me? I've done some modular arithmatic to determine factorability, but this loses me. Thanks.
Zurtex
#10
Dec19-04, 08:13 AM
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Quote Quote by mahesh_2961
hi ,
repeating an integer, x (say), six times is not the same as multiplying it six times ....
22715371505222715371505222715371505222715371505222715371505222715371505 2 is 227153715052 repeated 6 times and not (227153715052)6 and it is not a multiple of 7.
matt grime
#11
Dec19-04, 08:25 AM
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Either you or I have made a mistake, then, Zurtex. Now, if there truly were counter examples, then there would be a smaller on than the one you've just written. Incidentally, what is the remainder of that number mod 7?

As for dividing by 9.

suppose x/9 = y in the integers, then x=9y, and x=9y mod any integer, n say. If 9 and n are relativeyl prime then there is an integer k such that 9k=1 mod n thus xk=y mod n. 9 is a unit modulo 9, 9 is invertible. If 9 and 7 weren't coprime then I couldn't do this.
gonzo
#12
Dec19-04, 08:54 AM
P: 277
okay, I followed everything you said individually, but I don't understand "invertible".

I'm missing some piece of the puzzle here. Sorry for being dense. Maybe you could write it out in few steps starting from

s/9 * (10^6 -1)

I would start like this myself (which seems to be more complicated and possibly wrong):

10^6=10^3*10^3 and 10^3 mod 7 = -1

so the whole thing mod 7 = s/9(-1*-1 -1) = 0
shmoe
#13
Dec19-04, 09:41 AM
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Quote Quote by matt grime
Repeating a string, s, of r digits six times is the sam as the sum of a geometric progression. Let k be 10^r, it is

s +ks +k^2s+..+k^5s

The sum of this is (s/9)*(10^6-1)

since 9=2 mod 7, and thus invertible, and 10 is coprime with 7, this reduces to zero mod 7 quite trivially by fermat's little theorem.
Your geometric series should be s/(k-1)*(k^6-1)

k-1 won't be invertible mod 7 if r=0 mod 6 (Fermat's little theorem). Note Zurtex example has r=12 and Muzza has r=6. r=6 is the least number of digits you'll find for a counterexample.
matt grime
#14
Dec19-04, 10:06 AM
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ah, there we go, for some reason i'd got 9 rather than 10^r - 1.
Ethereal
#15
Dec19-04, 10:18 AM
P: n/a
I think mahesh really meant the following as he himself corrected:

Quote Quote by mahesh_2961
one correction to my query ... not any number but any integer
sorry for that
But in any case, what range of number string lengths can be repeated six times and be divisible by 7?
dextercioby
#16
Dec19-04, 10:26 AM
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Quote Quote by Ethereal
But in any case, what range of number string lengths can be repeated six times and be divisible by 7?
I believe Shmoe's post was pretty clear.Basically any natural number except natural multiples of 6.

Daniel.
gonzo
#17
Dec19-04, 10:33 AM
P: 277
Quote Quote by shmoe

k-1 won't be invertible mod 7 if r=0 mod 6 (Fermat's little theorem).
Is this because 10^6 mod 7 = 1 so that k-1=0?

Can someone give me a really simple definition of "invertible" here? Thanks.
Zurtex
#18
Dec19-04, 10:36 AM
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I've not been able to keep up the thread but I think you're in the wrong matt:

22715371505222715371505222715371505222715371505222715371505222715371505 2 mod 7 = 2

There will be smaller counter examples but that's the 1st one I found (mind you I only searched for about 10 secs.


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