# bases and binary

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

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.