Watch our videos of multiplication methods that you may not have met before. Can you make sense of them?

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

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" .

The number 10112359550561797752808988764044943820224719 is called a 'slippy number' because, when the last digit 9 is moved to the front, the new number produced is the slippy number multiplied by 9.

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

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

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.

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.

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

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. . . .

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?

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

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

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 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?

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

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

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

Number problems at primary level that may require resilience.

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?

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. . . .

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

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

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

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

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

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.

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

Alf describes how the Gattegno chart helped a class of 7-9 year olds gain an awareness of place value and of the inverse relationship between multiplication and division.

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

The Scot, John Napier, invented these strips about 400 years ago to help calculate multiplication and division. Can you work out how to use Napier's bones to find the answer to these multiplications?

Imagine you were given the chance to win some money... and imagine you had nothing to lose...

These pictures and answers leave the viewer with the problem "What is the Question". Can you give the question and how the answer follows?

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

Amy has a box containing domino pieces but she does not think it is a complete set. Which of her domino pieces are missing?

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

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.

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

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

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?

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?

More resources to support understanding multiplication and division through playing with numbers

Resources to support understanding of multiplication and division through playing with number.

Related resources supporting pupils' understanding of multiplication and division through playing with numbers.

In this article, Alf outlines six activities using the Gattegno chart, which help to develop understanding of place value, multiplication and division.

This article for primary teachers encourages exploration of two fundamental ideas, exchange and 'unitising', which will help children become more fluent when calculating.

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?

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