Question
A positive integer is called "Panache” if it can be written as a sum of distinct positive powers of 4, and "Elan" if it can be written as a sum of distinct positive powers of 6. From the given number, which of the following can we write as a sum of a "Panache” number and an "Elan" number?
Answer: Option D
:
D
Suppose that 2013 = a + b, where a is "Panache” and b is an "Elan". Then 'a' must be a sum of some of the powers of 4 less than 2013, namely 1, 4, 16, 64, 256, and 1024. Similarly, b must be a sum of some of the numbers 1, 6, 36, 216, and 1296. So, a + b must be a sum of some distinct entries from the list 1, 1, 4, 6, 16, 36, 64, 216, 256, 1024, 1296. If we use both 1024 and 1296, we get at least 1024+1296 = 2320 which is too big. But if we omit one of them, the most we can get 1 + 1 + 4 + 6 + 16 + 36 + 64 + 216 + 256 + 1296 = 1896; which is too small. So there is no way to achieve a sum of 2011, 2012 and 2013.
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:
D
Suppose that 2013 = a + b, where a is "Panache” and b is an "Elan". Then 'a' must be a sum of some of the powers of 4 less than 2013, namely 1, 4, 16, 64, 256, and 1024. Similarly, b must be a sum of some of the numbers 1, 6, 36, 216, and 1296. So, a + b must be a sum of some distinct entries from the list 1, 1, 4, 6, 16, 36, 64, 216, 256, 1024, 1296. If we use both 1024 and 1296, we get at least 1024+1296 = 2320 which is too big. But if we omit one of them, the most we can get 1 + 1 + 4 + 6 + 16 + 36 + 64 + 216 + 256 + 1296 = 1896; which is too small. So there is no way to achieve a sum of 2011, 2012 and 2013.
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