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

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

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

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

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

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?

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

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.

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

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

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

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.

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. . . .

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

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

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.

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). . . .

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.

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?

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?

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?

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

A game that tests your understanding of remainders.

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

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

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

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?

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

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

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?

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?

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

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

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

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

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

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

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!?

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?

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.

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

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

Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the. . . .

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.

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

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.