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        riddles >> easy >> Move the '6'
        
(Message started by: NickH on Oct 15^th, 2003, 12:11am)

Title: Move the '6'
Post by NickH on Oct 15^th, 2003, 12:11am

A positive integer ends in the (decimal) digit 6. If this 6 is moved to the beginning of the number (eg. 123456 becomes 612345), the resulting number is six times the original number.

What is the smallest original number that fits this condition?

Title: Re: Move the '6'
Post by BNC on Oct 15^th, 2003, 3:55am

::[hide]
Mark our number by N (N=??????6), and M=6N (M=6????????).

According to the rules of the riddle, looking from the right, the first digit of M is the 2nd of N, M's 2nd is N's 3rd, etc..

Multiplying N by 6, we'll get M. starting the process, we'll multiply N's rightmost 6 by 6 to get 36 => 6 and carry 3. The 6 is M's 1st digit, hence N's 2nd. Multiplying that 6 by 6, and adding the carry of 3 we'll get 39 => 9 and carry 3. The 9 is N's 3rd digit, thus we'll multiply it by 6 and add the carry of 3 to get 57 => 7 and carry 5. As we're calculating N and M simultaneously, continue until the conditions are fulfilled.

To ease the checking, it is easy to confirm that N's leftmost digits must be 101.

I used excel to calculate.

The result:

1016949152542372881355932203389830508474576271
1864406779661016949152542372881355932203389830
508474576271186440677966

[/hide]::

Title: Re: Move the '6'
Post by wowbagger on Oct 15^th, 2003, 4:17am

I did the same straightforward calculation as you, BNC, using pencil and paper. At N = ...71186440677966 I got tired and quit, however.

Title: Re: Move the '6'
Post by THUDandBLUNDER on Oct 15^th, 2003, 4:35am

Quote:

I used excel to calculate.

What a cop-out! ::) The Age of the Machines is upon us! :(

As 10 is a primitive root of 59, we can use the same method as in Centigrade to Fahrenheit (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_easy;action=display;num=1065349082).

Title: Re: Move the '6'
Post by wowbagger on Oct 15^th, 2003, 7:20am

on 10/15/03 at 04:35:39, THUDandBLUNDER wrote:

What a cop-out! ::) The Age of the Machines is upon us. :(

I wasn't exactly thrilled by BNC's solution either.

Quote:

As 10 is a primitive root of 59, we can use the same method as in Centigrade to Fahrenheit (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_easy;action=display;num=1065349082).

Looks like it's high time I catch up with most math-inclined people here regarding number theory. :-/
Or should I have a close look at these retrograde chess puzzles first? ???
[hide]No, T&B, I'm not really asking, so you don't have to respond.[/hide]

Title: Re: Move the '6'
Post by NickH on Oct 20^th, 2003, 3:13pm

10/59 = 0.1016949152542372881355932203389830508474576271186440677966 recurring.