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Question

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


A

>2

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B

1

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C

2

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D

none of these

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Solution

The correct option is A

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
n2(2a+n1)=128 (d=1)
n(2a+n1)=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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