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 non-negative 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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