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

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?

Mod 7

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

Double Time

Age 16 to 18 Challenge Level:
This is another exercise in re-arranging formulae involving modulus arithmetic and it illustrates another type of coding system. Alternatively the pairs $(\alpha, \beta)$ can also be treated as vectors and matrix algebra used to encode and decode the messages. Another alternative is to use an encryption formula based on the numbers 0 to 675 which represent the letter pairs.