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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.

Factorising with Multilink

Age 14 to 16
Challenge Level

Start with families of examples such as:

$x^2 + 3x + 2$
$x^2 + 4x + 3$
$x^2 + 5x + 4$


What do you notice?

Can you explain why?