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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:

Take each pair. Apply the decoding formulae. You will need to find the equivalent of 1/5, that is the multiplicative inverse of 5 modulo 26 (the number by which 5 is multiplied to give 1 modulo 26). Finally arrange the 44 letters into words putting the gaps in the appropriate places.