Modular arithmetic

problem
Favourite

Take Three From Five

Age

11 to 16

Challenge level

Caroline and James pick sets of five numbers. Charlie tries to find three that add together to make a multiple of three. Can they stop him?
problem
Favourite

Filling the Gaps

Age

14 to 16

Challenge level

Which numbers can we write as a sum of square numbers?
problem
Favourite

Guesswork

Age

14 to 16

Challenge level

Ask a friend to choose a number between 1 and 63. By identifying which of the six cards contains the number they are thinking of it is easy to tell them what the number is.
problem
Favourite

Prime AP

Age

16 to 18

Challenge level

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

Purr-Fection

Age

16 to 18

Challenge level

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

Old Nuts

Age

16 to 18

Challenge level

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?
problem

Mod 7

Age

16 to 18

Challenge level

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

Novemberish

Age

14 to 16

Challenge level

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.
problem

Euler's Officers

Age

14 to 16

Challenge level

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

Check Codes

Age

14 to 16

Challenge level

Details are given of how check codes are constructed (using modulus arithmetic for passports, bank accounts, credit cards, ISBN book numbers, and so on. A list of codes is given and you have to check if they are valid identification numbers?