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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.

Remainder Hunt

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

What are the possible remainders when the $100^{th}$ power of an integer is divided by $125$? To reduce the number of cases to be checked, express the number as $5p+q $ where $p $ and $q $ are integers and $q=1,2,3,4 $ and find the hundredth power using the Binomial Theorem.