Samia Henaine is a math teacher, and PYP and ICT Coordinator at Houssam Eddine Hariri High School in Saida (Lebanon). Her mission is to support teachers to build bridges by connecting concepts and forms of thinking within the mathematics classroom. Moreover, Samia encourages teachers to develop a growth mindset in relation to their students' mathematical potential.

Samia's website contains other inquiry prompts.

This activity from the Implementing the Mathematical Practice Standards Project requires students to look for evidence that might help them answer the question: Is there a general rule for how many different ways a number can be written as a sum of consecutive positive integers?

It turns out that there is a rule, which can be presented as a theorem: The number of ways a positive integer can be written as the sum of consecutive numbers is one fewer than the number of its odd factors.

The number of odd factors can be worked out using prime factorisation. For example, the prime factorisation of 18 = 2 x 3². Ignoring two, which will give even factors, add one to the exponent of 3 (that is, 2) and there are three (2 + 1) odd factors. Therefore, 18 can be written as the sum of consecutive numbers in two (3 - 1) ways (5 + 6 + 7 and 3 + 4 + 5 + 6).

In general, the number of odd factors can be calculated from the prime factorisation 2^p x a^q x b^r (a and b being odd) by finding the product of (q + 1)(r + 1). It follows that the number of ways of expressing a number as the sum of consecutive positive integers is (q + 1)(r + 1) - 1.

Examples

45 as the product of prime factors is 3² x 5. The number of the odd factors of 45 is (2 + 1)(1 + 1) = 6. Therefore, 45 can be expressed as the sum of consecutive positive integers in five (6 - 1) ways.

81 as the product of prime factors is 3⁴. The number of the odd factors of 81 is (4 + 1) = 5. Therefore, 81 can be expressed as the sum of consecutive positive integers in four (5 -1) ways.