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

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

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

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?

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

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). . . .

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.

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

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

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!?

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

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

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

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. . . .

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

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

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

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

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.

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?

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?

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

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

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

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

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

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?

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

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

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?

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

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?

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

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

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.

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

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

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?

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. . . .

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

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?

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

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

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.

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" .