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

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

A student in a maths class was trying to get some information from her teacher. She was given some clues and then the teacher ended by saying, "Well, how old are they?"

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

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" .

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?

This package contains a collection of problems from the NRICH website that could be suitable for students who have a good understanding of Factors and Multiples and who feel ready to take on some. . . .

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 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?

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

Is it possible to draw a 5-pointed star without taking your pencil off the paper? Is it possible to draw a 6-pointed star in the same way without taking your pen off?

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?

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

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

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

I am thinking of three sets of numbers less than 101. They are the red set, the green set and the blue set. Can you find all the numbers in the sets from these clues?

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. . . .

A game that tests your understanding of remainders.

This article for teachers describes how number arrays can be a useful reprentation for many number concepts.

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

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

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

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

Norrie sees two lights flash at the same time, then one of them flashes every 4th second, and the other flashes every 5th second. How many times do they flash together during a whole minute?

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

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?

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

Can you find a way to identify times tables after they have been shifted up?

Nearly all of us have made table patterns on hundred squares, that is 10 by 10 grids. This problem looks at the patterns on differently sized square grids.

A challenge that requires you to apply your knowledge of the properties of numbers. Can you fill all the squares on the board?

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

I am thinking of three sets of numbers less than 101. Can you find all the numbers in each set from these clues?

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

Factor track is not a race but a game of skill. The idea is to go round the track in as few moves as possible, keeping to the rules.

Follow this recipe for sieving numbers and see what interesting patterns emerge.

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?

Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice?

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

Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?

Becky created a number plumber which multiplies by 5 and subtracts 4. What do you notice about the numbers that it produces? Can you explain your findings?

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.

Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.

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?

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!?

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?

Can you work out what size grid you need to read our secret message?

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

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