Purr-fection

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.

In modulus arithmetic the only numbers involved are the whole numbers 0 to $m-1$ where $m$ is the modulus (or if you prefer it 1 to $m$). The multiplicative inverse of 7 (mod 26) is the number (equivalent to 1/7) that 7 is multiplied by to get the answer 1. The additive inverse of 17 is the number you add to 17 to get the answer 0 (mod 26).