Problem 22 · 2015 Math Kangaroo
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Number Theory
careful-countingnumber-systems
How many two-digit numbers can be written as sum of exactly six different powers of two? (Powers of two are \(2^0\), \(2^1\), \(2^2\), …)
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Answer: C — 2
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Hint 1 of 2
A sum of six different powers of two is a number with exactly six 1's in binary.
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Hint 2 of 2
Find two-digit numbers (10–99) whose binary form has exactly six ones.
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Approach: count two-digit numbers with six binary 1's
- Six different powers of two means six 1-bits in binary.
- Smallest such: 1 + 2 + 4 + 8 + 16 + 32 = 63 (= 111111 in binary).
- Swapping the 32 for 64 gives 1 + 2 + 4 + 8 + 16 + 64 = 95; any other choice exceeds 99.
- So there are 2 such numbers (63 and 95).
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