Number theory

problem
Really Mr. Bond

Age

14 to 16

Challenge level

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?
problem
A one in seven chance

Age

11 to 14

Challenge level

What is the remainder when 2^{164}is divided by 7?
problem
The public key

Age

16 to 18

Challenge level

Find 180 to the power 59 (mod 391) to crack the code. To find the secret number with a calculator we work with small numbers like 59 and 391 but very big numbers are used in the real world for this.
problem
Data chunks

Age

14 to 16

Challenge level

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.
problem
How much can we spend?

Age

11 to 14

Challenge level

A country has decided to have just two different coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?
problem
Where can we visit?

Age

11 to 14

Challenge level

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?
problem
Overlaps

Age

11 to 14

Challenge level

Can you find ways to put numbers in the overlaps so the rings have equal totals?
problem
Impossibilities

Age

11 to 14

Challenge level

Just because a problem is impossible doesn't mean it's difficult...
problem
Differences

Age

11 to 14

Challenge level

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

Age

14 to 16

Challenge level

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?