# powers of 2

• Nov 3rd 2008, 07:32 PM
nancymcwilliams
powers of 2
Every positive integer can be expressed in a unique way as the sum of powers of 2. Express 2008 as the sum of powers of 2.

• Nov 3rd 2008, 11:33 PM
ajj86
Possible solution
I don't know if this is what you were looking for, but I got an answer:

42^2 + 12^2 + 10^2 = 2008

I started by squaring 45 and then moving back from there with 44, 43 and so on. I then subtracted this squared value from 2008. When I got to 42^2, which equals 1764, I subtracted it from 2008 and got 244.

244 = 144 + 100
244 = 12^2 + 10^2

I then added these two squares to 42^2 and got 2008.
• Nov 3rd 2008, 11:53 PM
earboth
Quote:

Originally Posted by nancymcwilliams
Every positive integer can be expressed in a unique way as the sum of powers of 2. Express 2008 as the sum of powers of 2.

You are looking for a transcription of a decimal number into a dual number:

$2008=2^{10}+2^9+2^8+2^7+2^6+0\cdot 2^5+2^4+2^3+0\cdot 2^2+0\cdot 2^1+0\cdot 2^0$

Usually used writing:

$2008_{dec} = 11111011000_{bin}$