In how many ways 2^k (k12) can be written as sum of the two or more

Question

In how many ways can  2^k  (k>12) be written as sum of the two or more consecutive natural numbers?

A. 0
B. k-5
C. k-2
D. k
E.  k^2

nick1816 · Accepted Answer

Number of ways a number can be written as sum of the two or more consecutive natural numbers= Number of odd factors of number -1
Number of odd factors of 2^k= (0+1)= 1 {1 is the only odd factor}
Number of ways 2^k can be written as sum of the two or more consecutive natural numbers- 1-1=0

bv8562 · Answer

Any positive power to 2 will always produce an EVEN NUMBER and sum of two consecutive natural numbers is always ODD. Therefore, the number of ways  2^k  (k>12) can be written as sum of the two or more consecutive natural numbers is  ZERO (A)

druparel · Answer

but the sum of four consecutive numbers is even. so how can we assume there cannot be more than 1 way to sum the numbers?

pappal · Answer

since 2^13 and above can only be factorized as product of even numbers
eg: 2* 2^12 or 4*2^10 or 8*2^9 and so on
so neither of these even numbers can be a summation of consecutive numbers.
E+O->O, further this odd + next even no.->O
hence '0' no. of ways
hope its clear

EnglishAgast · Answer

Hello,

Is this a standard theory or an observation? Thank you.

bumpbot · Answer

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In how many ways 2^k (k>12) can be written as sum of the two or more

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