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?

I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?

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!?

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

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

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

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

Does this 'trick' for calculating multiples of 11 always work? Why or why not?

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" .

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

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

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

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

All the girls would like a puzzle each for Christmas and all the boys would like a book each. Solve the riddle to find out how many puzzles and books Santa left.

Your vessel, the Starship Diophantus, has become damaged in deep space. Can you use your knowledge of times tables and some lightning reflexes to survive?

If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why?

Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?

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?

When I type a sequence of letters my calculator gives the product of all the numbers in the corresponding memories. What numbers should I store so that when I type 'ONE' it returns 1, and when I type. . . .

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

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

What is the remainder when 2^{164}is divided by 7?

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

Investigate $1^n + 19^n + 20^n + 51^n + 57^n + 80^n + 82^n $ and $2^n + 12^n + 31^n + 40^n + 69^n + 71^n + 85^n$ for different values of n.

The value of the circle changes in each of the following problems. Can you discover its value in each problem?

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?

Can you find different ways of creating paths using these paving slabs?

Look on the back of any modern book and you will find an ISBN code. Take this code and calculate this sum in the way shown. Can you see what the answers always have in common?

Number problems at primary level that may require resilience.

Using the numbers 1, 2, 3, 4 and 5 once and only once, and the operations x and ÷ once and only once, what is the smallest whole number you can make?

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.

What is the largest number you can make using the three digits 2, 3 and 4 in any way you like, using any operations you like? You can only use each digit once.

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

This article for teachers looks at how teachers can use problems from the NRICH site to help them teach division.

This task combines spatial awareness with addition and multiplication.

This task offers an opportunity to explore all sorts of number relationships, but particularly multiplication.

Visitors to Earth from the distant planet of Zub-Zorna were amazed when they found out that when the digits in this multiplication were reversed, the answer was the same! Find a way to explain. . . .

Use your logical reasoning to work out how many cows and how many sheep there are in each field.

Amy has a box containing domino pieces but she does not think it is a complete set. She has 24 dominoes in her box and there are 125 spots on them altogether. Which of her domino pieces are missing?

Skippy and Anna are locked in a room in a large castle. The key to that room, and all the other rooms, is a number. The numbers are locked away in a problem. Can you help them to get out?

In November, Liz was interviewed for an article on a parents' website about learning times tables. Read the article here.

Find the smallest whole number which, when mutiplied by 7, gives a product consisting entirely of ones.

Using some or all of the operations of addition, subtraction, multiplication and division and using the digits 3, 3, 8 and 8 each once and only once make an expression equal to 24.

Amazing as it may seem the three fives remaining in the following `skeleton' are sufficient to reconstruct the entire long division sum.

What is the least square number which commences with six two's?

A 3 digit number is multiplied by a 2 digit number and the calculation is written out as shown with a digit in place of each of the *'s. Complete the whole multiplication sum.

Each clue in this Sudoku is the product of the two numbers in adjacent cells.

Play this game and see if you can figure out the computer's chosen number.

Look at what happens when you take a number, square it and subtract your answer. What kind of number do you get? Can you prove it?