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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 92,337

Hi;

This one was posed in another thread and solved there by anonimnystefy.

Find the smallest natural number for which, if you move its last digit to the beginning, you get a number that is 6 times the original. Or prove that no such number can exist.

We will follow his fine idea and see how we can utilize a CAS to help.

Suppose there were a 3 digit number that was the answer then:

or

No insight yet on the problem so we try a 4 digit number.

One more a 5 digit number:

We see a pattern developing and this is an important part of experimental mathematics.

The pattern is there is always a multiple of 59 on the left and a predictable form of 10^k - 6 on the right.

We know that

We also know that a0 which is the front digit of the new number must be one of these {1,2,3,4,5,6,7,8,9}.

The whole thing now presents itself as an algorithm that leads to this pseudocode:

1) Begin k loop

2) h = 10^k - 6 / 59

3) Times h by 1,2,3,4,5,6,7,8,9 and see if the products are integers.

4) Yes they are, print k and stop

5) They are not go back to 1.

*Last edited by bobbym (2013-03-02 17:32:08)*

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Is k the number of digits?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 92,337

Hi;

The answers are 58 digits long, but k+1 is the number of digits.

**In mathematics, you don't understand things. You just get used to them.**

**I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.**

**Online**

Is k coming as 16?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 92,337

Hi;

You probably got that answer because of precision problems.

Your programming language is just not carrying enough digits. You need to have a language that has higher precision than 17 significant digits.

When I invented this problem I had no idea that such an answer would be necessary.

**In mathematics, you don't understand things. You just get used to them.**

**I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.**

**Online**

I never knew python is that bad

I will import decimal module and do again

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 92,337

Does python support big integer arithmetic?

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Ya good, Thank You I am getting 57

P.S: Defining numbers in Python isn't a problem at all. How big do you want your integer to be?

*Last edited by Agnishom (2013-03-02 20:54:36)*

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 92,337

That is correct, for the first part!

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Hmm... Let me see if I can find the number

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 92,337

There is a little trick, you will have to refer to the math in post #1.

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`1694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661`

`169491525423728813559322033898305084745762711864406779661`

Is that it?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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I am close but something is wrong

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 92,337

Hi;

The great part of doing this type of work is that you never need to ask that question. You always have an

infallible way to know. Did you check that number?

**In mathematics, you don't understand things. You just get used to them.**

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Answer:

A. 1694915254237288135593220338983050847457627118644067796601

B. Something is still wrong

But B is 116 digits long

*Last edited by Agnishom (2013-03-02 21:28:05)*

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 92,337

Hi;

Do not worry, we will sort it all out.

You remember from the math:

The a0 can take any value from 1 - 9. You only picked 1. The left has every digit except for a0. When you choose

a digit for a0 it goes on the back of the number.

*Last edited by bobbym (2013-03-02 21:30:57)*

**In mathematics, you don't understand things. You just get used to them.**

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you wrote:

3) Times h by 1,2,3,4,5,6,7,8,9 and see if the products are integers.

Isn't this number of times the value of a_0?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 92,337

Hi;

It is a0!

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(10^57 - 6)*1 / 59 is an Integer.

So why won't 1 serve as a0?

*Last edited by Agnishom (2013-03-02 22:37:14)*

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 92,337

It sure could but that does not mean it solves the problem. As a matter of fact that are 4 solutions in this area.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,822

My solution was with a0=6 and a0=9.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 92,337

There are 2 more solutions.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,822

I figured that out. And I think that I will be able to find them as soon as I look at my notebook.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 92,337

Once Agnishom figures it out then I can post another one.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,822

I have a feeling it will be monstrous.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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