20071203, 21:31  #1 
May 2004
New York City
5×7×11^{2} Posts 
Sum of Digits
What's the smallest even positive integer which,
when represented in each base 1 through 50, has an even sum of digits? (What about bases 1 through 100?) 
20071203, 21:51  #2 
Jun 2003
12116_{8} Posts 
How do you represent something in base 1? 0, 00, 000???

20071203, 22:01  #3 
Account Deleted
"Tim Sorbera"
Aug 2006
San Antonio, TX USA
10256_{8} Posts 
http://en.wikipedia.org/wiki/Unary_numeral_system
8 (base 10) is 11111111 (base 1). Often written as tick marks to count things as they happen, with separators every 5. Back to the question, since it must have an even sum of digits in base 1, it must be an even number (not really sure where to go from there...I just noticed this trivial thing and thought I'd point it out). 
20071204, 00:51  #4 
Aug 2002
Ann Arbor, MI
433 Posts 
13655684 for 150. I'll have the smallest one for bases 1100 in a few minutes.

20071204, 01:11  #5  
Jun 2003
2×23×113 Posts 
Quote:


20071204, 01:52  #6 
Aug 2002
Ann Arbor, MI
433 Posts 
Just because it doesn't have the same properties doesn't mean it can't fall under the same definition. If you just think of base b as being all words in b letters (give the b letters an ordering, and then order all words lexigraphically, and make a bijection to the integers), then this is the natural notion of what base 1 would be.

20071204, 02:26  #7  
Jun 2003
12116_{8} Posts 
Quote:
I agree that it is /some/ type of number system, just not a conventional positional number system that is worthy of the "base n" moniker. [ Ok, so I'm only half serious about this whole thing  maybe somebody can move all this philosophical musings to someplace else] 

20071204, 05:57  #8 
Aug 2002
Ann Arbor, MI
433 Posts 
It's all a matter of how you define things. If you use a more general definition, base 1 is a very natural concept. If you use the the definition which relies on special properties you only have when b>2 (your "real" number bases), it's no surprise it's not going to extend to the case where you don't have special properties.
[Stupid second question about bases 1100...taking way too long] 
20071204, 10:38  #9  
"Jacob"
Sep 2006
Brussels, Belgium
2^{2}·439 Posts 
Quote:
Jacob 

20071204, 13:08  #10 
Account Deleted
"Tim Sorbera"
Aug 2006
San Antonio, TX USA
2×5×7×61 Posts 
I think base 1 is the most natural base, as it's used for counting things, but it is only useful, IMO, with the natural set of numbers, {1, 2, 3, ...}.

20071204, 15:43  #11  
"Lucan"
Dec 2006
England
2×3×13×83 Posts 
Quote:
2 doesn't enter into binary, so why should 1 enter into unary? And "A" doesn't occur in denary (decimal). David Last fiddled with by davieddy on 20071204 at 15:45 

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