Modular arithmetic

What remainders do you get when square numbers are divided by 4?

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

How many different ways can you arrange the officers in a square?

What are the possible remainders when the 100-th power of an integer is divided by 125?

What can you say about the common difference of an AP where every term is prime?

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

Find the remainder when 3^{2001} is divided by 7.

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

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?