Question

Find the number of ways in which 128 can be written as a sum of two or more consecutive natural numbers.

• A. >2
• B. 1
• C. 2
• D. none of these

Answer 1

The correct option is D

none of these

The number of ways of expressing a number as a sum of two or more consecutive numbers = (No. of odd factors of that number -1)
Since 128 is a power of 2, it does not have any odd factor. This means that it cannot be written as a sum of two or more consecutive natural numbers.

Alternate Method:
Sum of the consecutive numbers = 128
$⟹\frac{n}{2}(2a+n-1)=128(\because d=1)$
$⟹n(2a+n-1)=256(wheren>2)$
Now, all factors of 256 are even except 1 (n=1 is not possible).
If n is even, then 'a' will never have an integral value. Hence, no such series of consecutive numbers is possible.