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.

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. 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?

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

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" .

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.

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?

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

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

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

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...

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

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

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.

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). . . .

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 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?

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?

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.

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

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. . . .

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.

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.

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?

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

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

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?

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?

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

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. . . .

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.

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.

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?

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. . . .

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

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?

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

Have you seen this way of doing multiplication ?

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

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

Explore the relationship between simple linear functions and their graphs.

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.