Factors and multiples

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?

What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?

Complete the magic square using the numbers 1 to 25 once each. Each row, column and diagonal adds up to 65.

Data is sent in chunks of two different sizes - a yellow chunk has 5 characters and a blue chunk has 9 characters. A data slot of size 31 cannot be exactly filled with a combination of yellow and blue chunks, explore what sizes near to 31 can, or cannot, be exactly filled.

Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?

The sum of the cubes of two numbers is 7163. What are these numbers?

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

What is the lowest number which always leaves a remainder of 1 when divided by each of the numbers from 2 to 10?

Find and explain a short and neat proof that 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

How can you use just one weighing to find out which box contains the lighter ten coins out of the ten boxes?