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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

Prove that for every right angled triangle which has sides with integer lengths:

  1. the area of the triangle is even and
  2. the length of one of the sides is divisible by 5.