Make a line of green and a line of yellow rods so that the lines differ in length by one (a white rod)

Follow this recipe for sieving numbers and see what interesting patterns emerge.

A number N is divisible by 10, 90, 98 and 882 but it is NOT divisible by 50 or 270 or 686 or 1764. It is also known that N is a factor of 9261000. What is N?

Can you work out what size grid you need to read our secret message?

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. . . .

Take any pair of numbers, say 9 and 14. Take the larger number, fourteen, and count up in 14s. Then divide each of those values by the 9, and look at the remainders.

6! = 6 x 5 x 4 x 3 x 2 x 1. The highest power of 2 that divides exactly into 6! is 4 since (6!) / (2^4 ) = 45. What is the highest power of two that divides exactly into 100!?

Twice a week I go swimming and swim the same number of lengths of the pool each time. As I swim, I count the lengths I've done so far, and make it into a fraction of the whole number of lengths I. . . .

How many zeros are there at the end of the number which is the product of first hundred positive integers?

Can you find what the last two digits of the number $4^{1999}$ are?

Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?

Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?

The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?

Find the highest power of 11 that will divide into 1000! exactly.

The sum of the first 'n' natural numbers is a 3 digit number in which all the digits are the same. How many numbers have been summed?

The number 8888...88M9999...99 is divisible by 7 and it starts with the digit 8 repeated 50 times and ends with the digit 9 repeated 50 times. What is the value of the digit M?

Have you seen this way of doing multiplication ?

You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.

How many integers between 1 and 1200 are NOT multiples of any of the numbers 2, 3 or 5?

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

Which pairs of cogs let the coloured tooth touch every tooth on the other cog? Which pairs do not let this happen? Why?

This article takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.

Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). . . .

Given the products of diagonally opposite cells - can you complete this Sudoku?

Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

Given the products of adjacent cells, can you complete this Sudoku?

I'm thinking of a number. When my number is divided by 5 the remainder is 4. When my number is divided by 3 the remainder is 2. Can you find my number?

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

The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.

How many numbers less than 1000 are NOT divisible by either: a) 2 or 5; or b) 2, 5 or 7?

Find the number which has 8 divisors, such that the product of the divisors is 331776.

Substitution and Transposition all in one! How fiendish can these codes get?

What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A

Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.

Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?

Using your knowledge of the properties of numbers, can you fill all the squares on the board?

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

A game in which players take it in turns to choose a number. Can you block your opponent?

Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?

I put eggs into a basket in groups of 7 and noticed that I could easily have divided them into piles of 2, 3, 4, 5 or 6 and always have one left over. How many eggs were in the basket?

In how many ways can the number 1 000 000 be expressed as the product of three positive integers?

What is the smallest number of answers you need to reveal in order to work out the missing headers?

Complete the following expressions so that each one gives a four digit number as the product of two two digit numbers and uses the digits 1 to 8 once and only once.

What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.