Topic: Number of ways to express a number as the sum of two or more consecutive positive integers

I have a question for you: in how many ways a natural number N can be written as the sum of two or more consecutive natural numbers?

Well, this is a question type you are going to encounter frequently when you prepare for various entrance exams and in this article, with the help of a handy CAT Number System trick, I am going to explain to you how to solve this particular problem type.

All set to learn this magical CAT Number System trick then? Let’s get started.

Learn through an example:

This concept is best illustrated through an example. Let’s take up a sample problem.

Sample Problem: In how many ways 35 can be written as the sum of two or more consecutive natural numbers?

We can write 35 = 17 + 18 or 5 + 6 + 7 + 8 + 9 or 2 + 3 + 4 + 5 + 6 + 7 + 8.

In fact, there is no other way to write down 35 as the sum of consecutive natural numbers. There are only these three ways. But how do we arrive at this answer during the exam? Is there any CAT number system trick that can be employed to arrive at this answer and that too quickly? Yes, there is. Let’s learn it.

Let us try to write 35 as a sum of integers. These integers can single, zero, positive or negative. We have the following table:

NumbersTotal NumbersNumbersTotal Numbers
35Odd – 34 – 33- ……. +33 +34 + 35Even
17 + 18Even-16 – 15- ……. +15 + 16 + 17 +18Odd
5 + 6 + 7 + 8 + 9Odd-4 – 3 – …… + 3 + 4 + 5 + 6 + 7 + 8 + 9Even
2 + 3 + 4 + 5 + 6 + 7 + 8Odd– 1 + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8Even

 

Now in the above table, the first column (i.e. half the solutions) has all the positive integers whereas the second column (i.e. the other half of the solutions) has zero, positive and negative integers. Also, there are exactly 4 ways where the number of terms is even and exactly 4 ways where the number of terms is odd. It means that the number of solutions with positive terms is same as the number of solutions with an odd number of terms.

So, in order to find the total number of solutions, we just need to check the number of solutions with an odd number of terms.

Let there be x consecutive integers, where x is odd, whose sum is N. The average of these x integers will be the middle term given by N/x (N/x will be an integer here). It means that if x is odd and N/x is an integer, then the sum of x consecutive integers with average N/x is equal to N. Since x is odd here and it divides N, so we have to find the factors of N which are odd. Hence, the number of ways to write a number N as the sum of consecutive natural numbers is same as the number of odd factors of N.

But there is a catch. The odd factors of N also include 1. Now, it is senseless to write N as the sum of only one natural number N. So. we ignore the factor 1 and rewrite the above result as

“The number of ways to write a number N as the sum of two or more consecutive natural numbers is same as the (number of odd factors of N – 1).

Let us take up some more examples and practice what we have learnt. Example Problem 1: In how many ways 100 can be written as the sum of two or more consecutive natural numbers?

Solution: We have 100 = 22 × 52

Now to find the odd number of factors, just ignore 2 and the powers of 2 as multiplying with 2 will make the factor even. So we are left with 52 only. Hence the number of odd factors are 2 + 1 = 3 (the odd factors of 100 are 1, 5 and 25). It means that the total number of ways to write 100 as the sum of two or more consecutive natural numbers is 3 – 1 = 2.

Example Problem 2: In how many ways 225 can be written as the sum of two or more consecutive natural numbers?

Solution: We have 225 = 32 × 52. The total factors of 225 are (2 + 1) × (2 + 1) = 3 × 3 = 9

Note that there are no even factors of 225 as the number itself is odd. So, the total ways of writing 225 as the sum of two or more consecutive positive integers is 9 – 1 = 8.

Example Problem 3: In how many ways 128 can be written as the sum of two or more consecutive natural numbers?

Solution: Since 128 = 27 has no odd prime number in its factorization, so it has no odd factor except 1. So 128 cannot be written as the sum of two or more consecutive natural numbers.

Hope, with the help of the above example, this CAT Number System trick is absolutely clear to you and you can use this in various exam questions.