Question

In how many ways can the number 100 be written as a sum of two or more consecutive positive integers?

• A. 2
• B. 3
• C. 5
• D. 6

Answer 1

The correct option is A

2

Conventional Approach

To write a number as a sum of two or more consecutive positive integers, we need to form an AP with "n” terms with a common difference 1

Sum to "n” terms of an AP= $\frac{n}{2}$(2a + (n - 1)d) where d=1 (as the numbers are consecutive) n(2a+(n-1))=200

thus, we need to write 200 as a product of 2 numbers, where one number is odd and the other is even

this can be done in 2 ways

200= 5$\times$40

200=25$\times$8

Thus 100 can be expressed in 2 ways as a sum of two or more consecutive positive integers

Shortcut:-

The answer is just one step. The number of ways of writing any number as a sum of two or more

consecutive positive integers= number of odd factors of that number-1

100 = 2${}^2$$\times$5${}^2$ Number of odd factors=3

Number of ways of expressing 100 as a sum of two or more consecutive positive integers= 3 - 1=2