Double and half

anna_gg · 2012-12-02 01:55:19

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)

bobbym · 2012-12-02 03:16:57

Hi anna_gg;

anna_gg · 2012-12-02 03:19:55

bobbym wrote:

Hi anna_gg;

Bobbym,
Sorry, I had made a mistake - please read the new description!

bobbym · 2012-12-02 03:46:58

Hi;

anna_gg · 2012-12-02 05:34:34

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

bobbym · 2012-12-02 12:28:42

Hi;

There are more solutions.

scientia · 2012-12-02 16:51:02

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)

anna_gg · 2013-02-26 01:08:24

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.

scientia · 2013-02-26 12:25:03

anna_gg · 2013-02-28 04:39:16

scientia wrote:

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

phrontister · 2013-03-01 03:06:52

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)

anna_gg · 2013-03-05 08:57:44

Excellent! Thanks!

Nehushtan · 2013-03-16 11:54:06

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:

Math Is Fun Forum

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

Double and half

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

Re: Double and half

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

Re: Double and half

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

Re: Double and half

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

Re: Double and half

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

Re: Double and half

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

Re: Double and half

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

Re: Double and half

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

Re: Double and half

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

Re: Double and half

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

Re: Double and half

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

Re: Double and half

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

Re: Double and half

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