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

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?

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!?

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

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

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?

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

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

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

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

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

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" .

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.

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

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

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

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?

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

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. . . .

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. . . .

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?

A game that tests your understanding of remainders.

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

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

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

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

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?"

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?

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.

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

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

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

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some. . . .

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

The clues for this Sudoku are the product of the numbers in adjacent squares.

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.

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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?

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?

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

Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?

Given the products of diagonally opposite cells - can you complete this Sudoku?

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

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?

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

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?