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I can not figure this out. I started off by making a table and putting the base(2) on one side, the number in the middle and the remainder in the third row and for 2010 I got 11111011010 but I cant find the next two numbers please help and please explain in detail. Thank you =)

  1. to express 2010 in base 2, we use exactly 8 ones and 3 zeros. find the next two integers in base 2 that use the same number of ones and zeros
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Well, express your string in standard base-10, and see how far you are from 2010: 11111011010=0.1+1.2+0.4+...... how far is this from 2010 (i.e., what is the difference between 2010 and your string?).The general method is the same as for decimal expansions.Does that help? – gary Apr 19 '11 at 23:57
up vote 2 down vote accepted

Just shift the $1$ that is on the far right to the left and will you have the next integer in base 2 with the same number of ones and zeroes. (Think about what it means to shift the one to the left and you'll see why this is the next number)

Edit: An example: lets look at the binary number $101$. The binary base notation means that this number is equal to $1*2^2+0*2^1+1*2^0=5$ in the decimal base notation. So, if we want the next integer with the same numbers of ones and zeros, we would shift the $1$ at the far right one space to the left and we would get $110$, which is $1*2^2+1*2^1+0*2^0=6$ in decimal base. The same logic applies to your case.

To find the other one, you'll have to think a bit, but the process is the same.

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im sorry, im not following. if you move the one over ie (11111011010 becomes 11111011100) how would you figure out which integer creates that? – user8051 Apr 19 '11 at 23:50
No problem. I'll put a small example in the answer. – Leonardo Fontoura Apr 19 '11 at 23:52
ok thank you =) – user8051 Apr 19 '11 at 23:56
There you go, hope this is clearer. – Leonardo Fontoura Apr 19 '11 at 23:57
Yes thank you so much! =D – user8051 Apr 20 '11 at 0:00

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