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How many noughts are at the end of these giant numbers?

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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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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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Age 14 to 16 Challenge Level:

What is the largest number which, when divided into each of $1905$, $2587$, $3951$, $7020$ and $8725$, leaves the same remainder each time?

Modular arithmetic. Mathematical reasoning & proof. Divisibility. Expanding and factorising quadratics. Multiplication & division. Indices. Creating and manipulating expressions and formulae. Prime factors. Factors and multiples. Mathematical induction.