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.

Weekly Challenge 8: Sixinit

Stage: 5 Short Challenge Level:

Choose any whole number $n$ and calculate $n^3 + 11n$.

Is the answer a multiple of 6 for all choices of $n$?

The challenge is to explain why or why not.
Did you know ... ?
Modular arithmetic is widely used to verify automatically whether a number has been correctly entered to a system. Each identification number (e.g. for passports, bank accounts, credit cards, ISBN book numbers and so on) obeys a rule which makes it easy to check (most of the time) whether or not the number has been copied correctly. For this reason such numbers are also called check codes.