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From: Chris on 17 Aug 2005 04:39
Perhaps somebody knows the solution of or a reference for the following
problem.

A number G of n digits has the property that the number formed by the first
k digits is divisible by k, for k = 1,2, ... , n.

Find such a number G with a maximal number of digits.


Thanks for any help!

C. Wildhagen
Rotterdam
The Netherlands


From: Philippe 92 on 17 Aug 2005 05:05
Chris wrote :
> Perhaps somebody knows the solution of or a reference for the following
> problem.
>
> A number G of n digits has the property that the number formed by the first k
> digits is divisible by k, for k = 1,2, ... , n.
>
> Find such a number G with a maximal number of digits.
>
>
> Thanks for any help!
>
> C. Wildhagen
> Rotterdam
> The Netherlands

3608528850368400786036725 is the largest number with such a property.

Of course any substring from the left of this has the same property.

Let N be a number with such a property.
to be searched a digit d such that 10N+d has the property.

I call a "terminal" number a number N that can't be "expanded"
that is for which no digit d exists with 10N+d having the property.
The only "number" which can indefinitely be expanded is
0000...

All solutions are then given by the set of these "terminal" numbers.
This set is finite and there are only 2492 terminal numbers.

Interesting is a number with all digits differents.
The only solution is 3816547290

A study of this in my homepage (in french) at
http://chephip.free.fr/ie/sol012.html
(redirects automatically if other browsers that IE)

Regards.

--
philippe
mail : chephip at free dot fr
site : http://chephip.free.fr/

From: Don Reble on 17 Aug 2005 20:54
> A number G of n digits has the property that the number formed by the
> first k digits is divisible by k, for k = 1,2, ... , n.
>
> Find such a number G with a maximal number of digits.

> 3608528850368400786036725 is the largest number with such a property.

.... for base ten. Y'know, there may be a largest number in any base.
(The large bases are hard to check, though.) Here's what I get.

2 2
3 200220
4 2220301
5 4022042200
6 52323000003
7 222255000033402606
8 56147660642432202
9 8831805708264668401762
10 3608528850368400786036725
11 594028289306446AA604A08202
12 606890346850BA6800B036206464
13 420A26CC95820CCCA8B346A80062A0996C3
14 BA280A76B2D29008B63A784800604A1A6476CC0
15 6A99598095665B0EAC8A95000C9960B50AAC42
16 FAE06678C2E884607EB8B4E0B0A0F0603420342
17 97A6391BAEDB352E0AFB88A2402264C280AAB531D326D
18 AE6GA07A9EA0423220GGF6G0E492GC6400G0D408308006EC
19 F7BBHBC600AE4C2E1F0219G8AI807HE0D14GG055220CCC6I
20 HG8G546820C0060G1C4020600IC0AAG0B60CIC00GAG8AIE8F0CC

--
Don Reble djr(a)nk.ca
From: Gerry Myerson on 17 Aug 2005 22:17
In article <M2RMe.7784$p5.7592(a)nnrp.ca.mci.com!nnrp1.uunet.ca>,
Don Reble <djr(a)nk.ca> wrote:

> > A number G of n digits has the property that the number formed by the
> > first k digits is divisible by k, for k = 1,2, ... , n.
> >
> > Find such a number G with a maximal number of digits.
>
> > 3608528850368400786036725 is the largest number with such a property.
>
> ... for base ten. Y'know, there may be a largest number in any base.
> (The large bases are hard to check, though.) Here's what I get.
>
> 2 2
> 3 200220
> 4 2220301
> 5 4022042200
> 6 52323000003
> 7 222255000033402606
> 8 56147660642432202
> 9 8831805708264668401762
> 10 3608528850368400786036725
> 11 594028289306446AA604A08202
> 12 606890346850BA6800B036206464
> 13 420A26CC95820CCCA8B346A80062A0996C3
> 14 BA280A76B2D29008B63A784800604A1A6476CC0
> 15 6A99598095665B0EAC8A95000C9960B50AAC42
> 16 FAE06678C2E884607EB8B4E0B0A0F0603420342
> 17 97A6391BAEDB352E0AFB88A2402264C280AAB531D326D
> 18 AE6GA07A9EA0423220GGF6G0E492GC6400G0D408308006EC
> 19 F7BBHBC600AE4C2E1F0219G8AI807HE0D14GG055220CCC6I
> 20 HG8G546820C0060G1C4020600IC0AAG0B60CIC00GAG8AIE8F0CC

see
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eismum.c
gi
Evidently Alec Mihailovs has gone out to base 23,
but only the number of "digits" is reported.
See also http://beta.mapleprimes.com/blog/alec/ponder_this

--
Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: Keith A. Lewis on 18 Aug 2005 14:02
Don Reble <djr(a)nk.ca> writes in article <M2RMe.7784$p5.7592(a)nnrp.ca.mci.com!nnrp1.uunet.ca> dated Wed, 17 Aug 2005 18:54:54 -0600:
>> A number G of n digits has the property that the number formed by the
>> first k digits is divisible by k, for k = 1,2, ... , n.
>>
>> Find such a number G with a maximal number of digits.
>
>> 3608528850368400786036725 is the largest number with such a property.
>
>.... for base ten. Y'know, there may be a largest number in any base.
>(The large bases are hard to check, though.) Here's what I get.
>
> 2 2

You mean 10, right? 2 is not a valid digit in base-2.

> 3 200220
> 4 2220301
> 5 4022042200
> 6 52323000003
> 7 222255000033402606
> 8 56147660642432202
> 9 8831805708264668401762
>10 3608528850368400786036725
>11 594028289306446AA604A08202
>12 606890346850BA6800B036206464
>13 420A26CC95820CCCA8B346A80062A0996C3
>14 BA280A76B2D29008B63A784800604A1A6476CC0
>15 6A99598095665B0EAC8A95000C9960B50AAC42
>16 FAE06678C2E884607EB8B4E0B0A0F0603420342
>17 97A6391BAEDB352E0AFB88A2402264C280AAB531D326D
>18 AE6GA07A9EA0423220GGF6G0E492GC6400G0D408308006EC
>19 F7BBHBC600AE4C2E1F0219G8AI807HE0D14GG055220CCC6I
>20 HG8G546820C0060G1C4020600IC0AAG0B60CIC00GAG8AIE8F0CC

It's trivial to prove that base N has at least an N-digit number.

From a probabilistic perspective, base N is likely to peter out when
k! > N^k. Another way of putting it is that the expected number of
k-digit solutions is N^k/k!.

Actually I think (N-1)N^(k-1)/k! would be more accurate, assuming leading
0's are disallowed...

--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.
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