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# Factorising with Multilink

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Age 14 to 16

Challenge Level

*For this problem, you will either need multilink cubes or these sets of number base sheets to cut out: Base Three Base FourBase Five Base Six.*

In the video below, Charlie and Becky show how you can make rectangles using sets of squares, sticks and units.

Can you make a rectangle to represent $x^2 + 7x + 12$?

Can you do it in more than one base?

Watch the video below to see how Charlie and Becky tackled this question:

**Take 1 square and 12 units in your chosen base. Put them together with some sticks to make rectangles that will work in all bases.**

*Charlie and Becky made $x^2 + 7x + 12$ into a rectangle with length $x+4$ and width $x+3$.*

How many different rectangles can you make?

What do you notice about the dimensions of your rectangles?

Imagine you had 1 square, lots of sticks and 100 units. What can you say about the dimensions of the rectangles it is possible to make?

**Now, take 1 square and 12 sticks in your chosen base. ****Put them together with some units to make rectangles that will work in all bases.**

How many different rectangles can you make?

What do you notice about the dimensions of your rectangles?

Imagine you had 1 square, 100 sticks and lots of units. What can you say about the dimensions of the rectangles it is possible to make?

**If you had 1 square, $p$ sticks and $q$ units, what can you say about the dimensions of the rectangles it is possible to make?**

**Extension**

Think about the rectangles it's possible to make if you use two, three, four... squares, some sticks and some units.

*You may also be interested in the other problems in our Getting started, getting stuck* *Feature*.

*Many thanks to Kenneth Ruthven and Paul Andrews whose ideas inspired this problem.*

Two cubes, each with integral side lengths, have a combined volume equal to the total of the lengths of their edges. How big are the cubes? [If you find a result by 'trial and error' you'll need to prove you have found all possible solutions.]

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.