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Purr-fection

What is the smallest perfect square that ends with the four digits 9009?

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Old Nuts

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?

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Mod 7

Find the remainder when 3^{2001} is divided by 7.

Pythagoras Mod 5

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Why do this problem?

To tackle this problem from first principles provides a real challenge. The solution published here was done by a 17 year old who also proved the second part from first principles.

The arguments in the proofs use modulus arithmetic in an informal way.

Possible approaches

Some students may like to read the articles on Pythagorean triples first and then try to use similar arguments to prove the results in this problem from first principles.

Another approach (and an easier one) would be to use the well known formula for generating Pythagorean triples (proved in the articles on Pythagorean triples) as the basis for a proof of the results in this problem.

Key question

If we can prove the result when $x^2 +y^2= z^2$ and $x, y$ and $z$ have no common factors (for primitive triples), are the results then true for all right angled triangles?