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

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?

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" .

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

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

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?

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.

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?

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?

These red, yellow and blue spinners were each spun 45 times in total. Can you work out which numbers are on each spinner?

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

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

Are these statements always true, sometimes true or never true?

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?

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

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

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?

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

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. . . .

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?

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

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?

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. . . .

Katie and Will have some balloons. Will's balloon burst at exactly the same size as Katie's at the beginning of a puff. How many puffs had Will done before his balloon burst?

How can you use just one weighing to find out which box contains the lighter ten coins out of the ten boxes?

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?

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

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

Andrew decorated 20 biscuits to take to a party. He lined them up and put icing on every second biscuit and different decorations on other biscuits. How many biscuits weren't decorated?

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

Find the words hidden inside each of the circles by counting around a certain number of spaces to find each letter in turn.

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

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

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

There are a number of coins on a table. One quarter of the coins show heads. If I turn over 2 coins, then one third show heads. How many coins are there altogether?

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

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.

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.

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!?

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?"

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

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

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?

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?

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

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

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