Quite Binary
Number Theory
Level
3
What is the sum (in base 10) of all the natural numbers less than 64 which have exactly two ones in their base 2 representation?
The answer is 315.
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1 solution
Since 6 4 = 2 6 , therefore all natural number, which are smaller than 64 can be represented with a maximum of 6 binary digits.
If we want to have a binary number with (maximum 6 digits and) exactly two "1" digits, we can consider a choice of 2 out of 6 binary places. We can choose (the first digit 6 ways, and) the second digit 5 ways. This means, that there are exactly 5 numbers, which share the same "1" digit at the same place (e.g. all have a "1" as the third digit, our first choice).
Therefore, the sum we are looking for:
5 × ( 3 2 + 1 6 + 8 + 4 + 2 + 1 ) = 5 × 6 3 = 3 1 5