Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

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?

I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

I added together the first 'n' positive integers and found that my answer was a 3 digit number in which all the digits were the same...

Can you explain the strategy for winning this game with any target?

Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

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?

Can you find any perfect numbers? Read this article to find out more...

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?

The clues for this Sudoku are the product of the numbers in adjacent squares.

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

Is there an efficient way to work out how many factors a large number has?

Can you find a way to identify times tables after they have been shifted up or down?

Can you find any two-digit numbers that satisfy all of these statements?

Got It game for an adult and child. How can you play so that you know you will always win?

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

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

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

Factor track is not a race but a game of skill. The idea is to go round the track in as few moves as possible, keeping to the rules.

The five digit number A679B, in base ten, is divisible by 72. What are the values of A and B?

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

Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 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.

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?

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

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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.

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

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. . . .

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

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

I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some. . . .

When the number x 1 x x x is multiplied by 417 this gives the answer 9 x x x 0 5 7. Find the missing digits, each of which is represented by an "x" .

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.

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

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

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

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

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

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?

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

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.