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Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

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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.

LCM Sudoku

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

You can express $8, 9, 6, 4, 2, 5$ and $7$ as $2^3, 3^2, 2 \times 3, 2^2, 2, 5$ and $7$.

That means the LCM must contain $2^3, 3^2, 5$ and $7$ to ensure that it is a multiple of all the numbers.

So the LCM is $2^3 \times 3^2 \times 5 \times 7 = 2520$.