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Two Cubes

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.]

Common Divisor

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.

Finding Factors

Age 14 to 16
Challenge Level

The interactivity below is similar to the one used in Missing Multipliers so you may wish to work on that first.

In the multiplication grid below, the headings and the answers have been hidden. Each of the headings is an expression of the form $x + a$ where $a$ is an integer between $-5$ and $5$. By revealing some of the answers, can you work out what each heading must be? 
Drag the green and purple labels onto the headers to make the correct expressions.

What is the smallest number of answers you need to reveal in order to work out the missing headers?

Once you've developed strategies for finding the factors in the interactivity above, you might like to have a go at solving some bigger grids which include coefficients of $x$ greater than 1 in the row and column headings:
Six by six grid
Eight by eight grid
Ten by ten grid

You may also be interested in the other problems in our Working backwards, leaping forwards Feature.