### 14 Divisors

What is the smallest number with exactly 14 divisors?

### Helen's Conjecture

Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?

### Oh! Hidden Inside?

Find the number which has 8 divisors, such that the product of the divisors is 331776.

# Summing Consecutive Numbers

### Why do this problem?

This problem offers a simple context to begin exploration that naturally leads to some interesting conjectures about properties of numbers, and the possibility of developing some quite sophisticated algebraic arguments and proofs.

### Possible approach

One approach is to introduce the lesson in the way it is introduced on the video, with students suggesting numbers and then the teacher at first, and then the whole class, finding ways to express them as consecutive sums.

Here is an alternative approach:
"Can someone give me a set of two or more consecutive numbers?"
Write a few sets on the board.

"What totals do we get by adding the consecutive numbers in these sets?"

Write + signs in between the lists of numbers.

"These totals are all examples of numbers that can be written as the sum of consecutive numbers. Do you think all numbers can be written in this way?"

"How about trying to write the numbers from $1$ to $30$ as sums of consecutive numbers?"

Give students time to work in pairs on filling in the gaps from $1$ to $30$. While they are working, write the numbers from $1$ to $30$ on the board ready to collect together the sums the class have found.

If students ask about negative numbers, one possible answer is: "Stick to positive numbers for now, and then perhaps investigate negative numbers later."

Once most pairs have filled in most of the gaps, collect their results on the board.

"Spend a minute looking at these results and then be prepared to talk about anything interesting that you notice."

Give them time to think on their own at first and then share ideas with their partner, before discussion with the whole class.

Next, collect together any noticings, and write them on the board in the form of questions or conjectures. If such conjectures are not forthcoming, there are some suggested lines of enquiry in the problem.

Allow pairs time to work on the conjectures of their choosing, reminding them that they will need to provide convincing arguments to explain any of their conclusions.

If appropriate, bring the class together to spend some time discussing algebraic representations of consecutive numbers ($n, n+1, n+2...$) to give students the tools to create algebraic proofs.

Finally, students could create a poster, a presentation or a short report explaining one key conclusion that they came to, together with the convincing arguments they used to explain it.

### Key questions

What happens when you add together a pair of consecutive numbers? Three consecutive numbers? Four?
What do you notice about the numbers you can't make?
If the first of a set of consecutive numbers is $n$, how can you express the next numbers in the set, and hence the total, algebraically?

### Possible extension

Challenge students to find an efficient way to calculate how many different ways $x$ can be expressed as a sum of consecutive numbers, for any $x$.

More challenging extension - prove that it is not possible to write $2^n$ as the sum of consecutive positive whole numbers, for any $n$.

Pair Products offers another good context for exploring properties of numbers and then using algebra to represent and explain the results.

### Possible support

When collecting together the class's results for the numbers from $1$ to $30$, they can be arranged on the board in ways that will make it easier for patterns to emerge. Charlie's results from the problem could be set out like this:

 9 = 4+5 2+3+4 10 = 1+2+3+4 11 = 5+6 12 = 3+4+5 13 = 6+7 14 = 2+3+4+5 15 = 7+8 4+5+6 1+2+3+4+5