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

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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.

Modular Fractions

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

$$3^{-1}+6^{-1} = 5 + 6 = 11 = 4 \pmod 7$$

and

$$(3+6)^{-1} = 9^{-1} = 2^{-1} = 4 \pmod 7$$

so $x=3,\ y=6$ is one solution. Now find the other solutions.