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

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

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...

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. . . .

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

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.

Explore the relationship between simple linear functions and their graphs.

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?

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

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

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

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

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

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?

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?

A collection of resources to support work on Factors and Multiples at Secondary level.

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). . . .

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?

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 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?

A game that tests your understanding of remainders.

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

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.

Find the frequency distribution for ordinary English, and use it to help you crack the code.

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

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

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

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

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?

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

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

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. . . .

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

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

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

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?

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?

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

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

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!?

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?

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.

Some 4 digit numbers can be written as the product of a 3 digit number and a 2 digit number using the digits 1 to 9 each once and only once. The number 4396 can be written as just such a product. Can. . . .

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

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

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" .

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

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