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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

An easy way to prove these results is to use the well known formula for generating Pythagorean triples and assume that all Pythagorean triples can be generated in the same way. The formula is proved and the ideas discussed at length in the articles Pythagorean Triples I, Pythagorean Triples II and Picturing Pythagorean Triples.

If you want more of a challenge you can try to prove the result from first principles.