Extraterrestrial numbers
Number Theory
Level
4
Every positive integer can be expressed as a sum of distinct powers of 2. Note that 1 and 2 are powers of 2. How many 3 digit positive integer is/are the sum of exactly 9 distinct powers of 2?
The answer is 5.
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1 solution
There are 10 powers of 2 less than 1,000 (we clearly can't use any that's more than 1,000, otherwise we couldn't have a 3-digit number.) 2 9 + 2 8 + 2 7 + 2 6 + 2 5 + 2 4 + 2 3 + 2 2 + 2 1 + 2 0 = 2 1 0 − 1 = 1023. We have to omit exactly one of those numbers to leave a sum less than 1,000. So the number we omit has to be greater than 23, which eliminates 5 of the 10 possibilities ( 2 4 , 2 3 , 2 2 , 2 1 a n d 2 0 ) .
That leaves 10-5 = 5 solutions.