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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.]

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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.

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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

Stage: 4 Challenge Level: Challenge Level:1

Patrick from Woodbridge School described the strategy he used to find the headings:

The method I used was to reveal all of one diagonal and half of the remaining diagonal. This is the minimal way of revealing two expressions which share each factor, and it just fits into the Level 3 rules. Factoring each expression then reveals a few factors for each, and we can build up possibilities by checking for shared factors down one column or row.