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 [SOLVED] powers of 2

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peyman
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PostSubject: [SOLVED] powers of 2   Thu Nov 19, 2009 7:12 pm

I there a power of 2 such that its digits could be rearranged and made into another power of 2? (No zeroes are allowed in the leading digit: for example. 0032 is not allowed.)
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Bruno
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PostSubject: Re: [SOLVED] powers of 2   Thu Nov 19, 2009 10:37 pm

No, because obviously the biggest of the two could be at most 8 times the other, and the difference between the two would be divisible by 9 (casting out nines!). But it's easy to check that none of

2^(m+3)-2^m,

2^(m+2)-2^m,

2^(m+1)-2^m

are divisible by 9.
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Bruno
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PostSubject: Re: [SOLVED] powers of 2   Thu Nov 19, 2009 10:39 pm

Now that was literally a 60-second problem! Mohammad would be proud. alien
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peyman
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PostSubject: Re: [SOLVED] powers of 2   Thu Nov 19, 2009 11:31 pm

A shorter answer would've been "No, obviously!"

Can you provid more details of your "obvious" claims?
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Bruno
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PostSubject: Re: [SOLVED] powers of 2   Fri Nov 20, 2009 9:11 am

Haha. Okay, sure :

Well the quotient between the greatest and the smallest is a power of 2. This quotient can't be a power of 2 greater than 8 or else the biggest of the two numbers would have more digits than the other. So the biggest of the two is either twice the other, four times the other, or eight times the other.

Moreover both numbers are congruent (mod 9) because every positive integer is congruent to the sum of its decimal digits (mod 9). So the difference between the two must be divisible by 9. But none of

2^(m+3)-2^m = 2^m x 7,

2^(m+2)-2^m = 2^m x 3,

2^(m+1)-2^m = 2^m

are divisible by 9.
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peyman
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PostSubject: Re: [SOLVED] powers of 2   Fri Nov 20, 2009 2:09 pm

very good! cheers

(you're starting to solve problems like mohammad, saying everything is obvious!)
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Bruno
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PostSubject: Re: [SOLVED] powers of 2   Fri Nov 20, 2009 2:17 pm

peyman wrote:

(you're starting to solve problems like mohammad, saying everything is obvious!)

Everything is obvious until proved otherwise, right? rabbit
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peyman
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PostSubject: Re: [SOLVED] powers of 2   Fri Nov 20, 2009 2:21 pm

obviously!
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Mohammad
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PostSubject: Re: [SOLVED] powers of 2   Fri Nov 20, 2009 2:46 pm

Bruno, you should start saying also "we can simply show". I used this sentence often in measure theory assignments and Prof Satncu writing me always "no wast of paper here" or "a bit of rush here" Very Happy , and I also got slapped from the guy was training us math competition in high school for saying "it is obvious". bounce
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PostSubject: Re: [SOLVED] powers of 2   Fri Nov 20, 2009 3:18 pm

Mohammad wrote:
Bruno, you should start saying also "we can simply show". I used this sentence often in measure theory assignments and Prof Satncu writing me always "no wast of paper here" or "a bit of rush here" Very Happy , and I also got slapped from the guy was training us math competition in high school for saying "it is obvious". bounce

Haha... I also like "it's easy to show/see that...".
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