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 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?

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

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

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?

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!?

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.

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.

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

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?

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?

56 406 is the product of two consecutive numbers. What are these two numbers?

Is there an efficient way to work out how many factors a large number has?

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

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?

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?

Number problems at primary level that may require determination.

Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?

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

This big box multiplies anything that goes inside it by the same number. If you know the numbers that come out, what multiplication might be going on in the box?

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

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

What is the lowest number which always leaves a remainder of 1 when divided by each of the numbers from 2 to 10?

Work out Tom's number from the answers he gives his friend. He will only answer 'yes' or 'no'.

On the planet Vuv there are two sorts of creatures. The Zios have 3 legs and the Zepts have 7 legs. The great planetary explorer Nico counted 52 legs. How many Zios and how many Zepts were there?

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

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

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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

Number problems at primary level to work on with others.

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.

Have a go at balancing this equation. Can you find different ways of doing it?

Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it?

Can you work out some different ways to balance this equation?

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

A game that tests your understanding of remainders.

Can you see how these factor-multiple chains work? Find the chain which contains the smallest possible numbers. How about the largest possible numbers?

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 this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?

Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on?

"Ip dip sky blue! Who's 'it'? It's you!" Where would you position yourself so that you are 'it' if there are two players? Three players ...?

When Charlie asked his grandmother how old she is, he didn't get a straightforward reply! Can you work out how old she is?

Explore the relationship between simple linear functions and their graphs.

Given the products of adjacent cells, can you complete this Sudoku?

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. . . .

Benâ€™s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

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