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Topic: Move the '6' (Read 443 times) |
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NickH
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Move the '6'
« on: Oct 15th, 2003, 12:11am » |
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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?
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BNC
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Re: Move the '6'
« Reply #1 on: Oct 15th, 2003, 3:55am » |
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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
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| « Last Edit: Oct 15th, 2003, 3:56am by BNC » |
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How about supercalifragilisticexpialidociouspuzzler [Towr, 2007]
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wowbagger
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Re: Move the '6'
« Reply #2 on: Oct 15th, 2003, 4:17am » |
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I did the same straightforward calculation as you, BNC, using pencil and paper. At N = ...71186440677966 I got tired and quit, however.
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| « Last Edit: Oct 15th, 2003, 4:19am by wowbagger » |
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"You're a jerk, <your surname>!"
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ThudnBlunder
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Re: Move the '6'
« Reply #3 on: Oct 15th, 2003, 4:35am » |
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Quote:| I used excel to calculate. |
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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.
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| « Last Edit: Oct 15th, 2003, 12:23pm by ThudnBlunder » |
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THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
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wowbagger
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Re: Move the '6'
« Reply #4 on: Oct 15th, 2003, 7:20am » |
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on Oct 15th, 2003, 4:35am, THUDandBLUNDER wrote:
What a cop-out! The Age of the Machines is upon us. |
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I wasn't exactly thrilled by BNC's solution either.
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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? 
No, T&B, I'm not really asking, so you don't have to respond.
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| « Last Edit: Oct 15th, 2003, 7:21am by wowbagger » |
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NickH
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Re: Move the '6'
« Reply #5 on: Oct 20th, 2003, 3:13pm » |
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10/59 = 0.1016949152542372881355932203389830508474576271186440677966 recurring.
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