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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.

The Public Key

Age 16 to 18 Challenge Level:
This problem is intended to be tackled in conjunction with reading the article on Public Key Cryptography though it is not essential to understand the details of Public Key Cryptography in order to do the problem. In itself it is a challenge to systematically reduce the large number $180^{59}$ (using modulus arithmetic) to one which the calculator can handle and finally to its equivalent modulo 391.