Question
Find the number of ways in which 128 can be written as a sum of two or more consecutive natural numbers.
Answer: Option D
:
D
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
⟹n2(2a+n−1)=128 (∵d=1)
⟹n(2a+n−1)=256 (where n>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.
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:
D
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
⟹n2(2a+n−1)=128 (∵d=1)
⟹n(2a+n−1)=256 (where n>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.
Was this answer helpful ?
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