Posted by Anonymous on Sunday, June 23, 2013 at 8:57am.
How many ways are there to write 2000 as the sum of powers of 2 with nonnegative exponents, where each power is used at most twice?

math  MathMate, Sunday, June 23, 2013 at 10:09am
Divide by two until the remainder is 0 or 1.
The corresponding quotients will give the powers of 2 to make up 2000.
2000/2=1000 R0
1000/2=500 R0
500/2=250 R0
250/2=125 R0
125/2=62 R1
62/2=31 R0
31/2=15 R1
15/2=7 R1
7/2=3 R1
3/2=1 R1
So 2000 in binary notation is
11111010000
or
the powers are
4,6,7,8,9,10,11
or
2^4+2^6+2^7+2^8+2^9+2^10
=2000
There is only one way if the powers are not to be repeated.
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