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### A Useful Property For The Powers Of 2

Powers of 2 are very helpful in calculations. Candidates should memorise powers of 2 upto 12 so that it can be used in the questions.

 20 1 21 2 22 4 23 8 24 16 25 32 26 64 27 128 28 256 29 512 210 1024 211 2048 212 4096
1. The sum of powers of 2 from 0 to any number n will be equal to 2n+1 – 1.
If a number is written from 1 to N as a sum of one or more of the integers of a given set of integers, then it can easily be done from powers of 2. The set of integers used by us comprise of all the powers of 2 starting form 1 (i.e. 20) to the largest power of 2 less than or equal to N.

#### For example:

If you want to build all the integers upto 255, the numbers 1, 2, 4, 8, 16, 32, 64, 128 are sufficient as 255=1+2+4+8+16+32+64+128.
1. Differently, if we have one weight each of 1, 2, 4, 8, 16, 32, 64 and 128 kg, then all the items would be measured from 1 kg to 255 kg using one or more of the given weights (the weights used only in one pan of the weighing scales).

#### Example:

How much minimum number of weights are required to weigh all possible weights upto 512 Kg (Putting all the weights only in one side of pan)

#### Solution:

512=29. Minimum Number of weights required=9+1=10. The weights will be 1,2, 4, 8, 16, 32, 64,128, 256 kg