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## #1 2012-12-02 01:55:19

anna_gg
Member
Registered: 2012-01-10
Posts: 232

### Double and half

Find a natural number for which, if you move its first digit at the end, you get a number that is half the original one
(e.g. 81345--->13458 but the resulting number must be the half of the original).

Last edited by anna_gg (2012-12-02 03:16:06)

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## #2 2012-12-02 03:16:57

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Double and half

Hi anna_gg;

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #3 2012-12-02 03:19:55

anna_gg
Member
Registered: 2012-01-10
Posts: 232

bobbym wrote:

Hi anna_gg;

Bobbym,

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## #4 2012-12-02 03:46:58

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Double and half

Hi;

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #5 2012-12-02 05:34:34

anna_gg
Member
Registered: 2012-01-10
Posts: 232

### Re: Double and half

bobbym wrote:

Hi;

Great! It is said that there are infinitely many solutions. I have used Excel for the calculations but have not been able to find any at the range 1-5,000,000. Then I gave up

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## #6 2012-12-02 12:28:42

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Double and half

Hi;

There are more solutions.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #7 2012-12-02 16:51:02

scientia
Member
Registered: 2009-11-13
Posts: 224

### Re: Double and half

Suppose the number is

NB:
(i) Since the number must be even, I write

for the last digit;
.

(ii)

and
cannot be zero but the other
can be 0.

So we want

i.e.

As the LHS is divisible by 19, so must the RHS, and as

we must have
divisible by 19. The smallest such n is
 in other words, the smallest solution is an 18-digit number.

PS: Yes, there are infinitely many solutions, because there are infinitely many n such that

is divisible by 19.

Last edited by scientia (2012-12-02 16:54:02)

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## #8 2013-02-26 01:08:24

anna_gg
Member
Registered: 2012-01-10
Posts: 232

### Re: Double and half

Here is another variation: Find the smallest natural number for which, if you move its last digit at the beginning, you get a number that is 5 times the original.

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## #9 2013-02-26 12:25:03

scientia
Member
Registered: 2009-11-13
Posts: 224

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## #10 2013-02-28 04:39:16

anna_gg
Member
Registered: 2012-01-10
Posts: 232

### Re: Double and half

scientia wrote:

CORRECT! I found it by using Excel but now am trying to formulate it.

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## #11 2013-03-01 03:06:52

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,592

### Re: Double and half

Hi anna_gg,

Try the following in Excel (I use v2007):-

And here's a little program in BASIC, but testing more numbers, with the same single result (142857):

FOR n = 10 TO 5000000000 STEP 5
n\$ = STR\$(n)
a\$ = RIGHT\$(n\$,LEN(n\$)-1) + LEFT\$(n\$,1)
IF VAL(a\$)*5 - n = 0 THEN PRINT a\$
NEXT n
END

Last edited by phrontister (2013-03-04 00:40:44)

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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## #12 2013-03-05 08:57:44

anna_gg
Member
Registered: 2012-01-10
Posts: 232

### Re: Double and half

Excellent! Thanks!

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## #13 2013-03-16 11:54:06

Nehushtan
Member
Registered: 2013-03-09
Posts: 896
Website

### Re: Double and half

So, we generalize the problem as follows:

Find the smallest natural number of two or more digits such that if you move the last digit to the front, the resulting number will be n times the original number.

Here are my calculations:

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