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

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

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.

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

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?

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?

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). . . .

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

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

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

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

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?

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. . . .

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...

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

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

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

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

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

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. . . .

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.

Explore the factors of the numbers which are written as 10101 in different number bases. Prove that the numbers 10201, 11011 and 10101 are composite in any base.

Using the digits 1, 2, 3, 4, 5, 6, 7 and 8, mulitply a two two digit numbers are multiplied to give a four digit number, so that the expression is correct. How many different solutions can you find?

Each letter represents a different positive digit AHHAAH / JOKE = HA What are the values of each of the letters?

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?

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

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.

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

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

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?

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. . . .

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?

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.

What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?

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

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.

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

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

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?

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

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

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

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

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

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

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