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.

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

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

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

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?

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

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?

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

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

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" .

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?

This problem is designed to help children to learn, and to use, the two and three times tables.

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?

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

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.

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!?

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

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 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.

After training hard, these two children have improved their results. Can you work out the length or height of their first jumps?

Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?

Here is a picnic that Petros and Michael are going to share equally. Can you tell us what each of them will have?

A game for 2 people using a pack of cards Turn over 2 cards and try to make an odd number or a multiple of 3.

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

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

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. . . .

There are four equal weights on one side of the scale and an apple on the other side. What can you say that is true about the apple and the weights from the picture?

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?

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

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.

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?

Take the number 6 469 693 230 and divide it by the first ten prime numbers and you'll find the most beautiful, most magic of all numbers. What is it?

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.

Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

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?

This challenge is a game for two players. Choose two numbers from the grid and multiply or divide, then mark your answer on the number line. Can you get four in a row before your partner?

Use this information to work out whether the front or back wheel of this bicycle gets more wear and tear.

The clockmaker's wife cut up his birthday cake to look like a clock face. Can you work out who received each piece?

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?

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?

Use 4 four times with simple operations so that you get the answer 12. Can you make 15, 16 and 17 too?

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

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?

Bernard Bagnall recommends some primary school problems which use numbers from the environment around us, from clocks to house numbers.

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?

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?

Grandma found her pie balanced on the scale with two weights and a quarter of a pie. So how heavy was each pie?

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

Start by putting one million (1 000 000) into the display of your calculator. Can you reduce this to 7 using just the 7 key and add, subtract, multiply, divide and equals as many times as you like?