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 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.

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

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. . . .

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?

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

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

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

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?

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!?

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

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

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 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?

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 mathematician goes into a supermarket and buys four items. Using a calculator she multiplies the cost instead of adding them. How can her answer be the same as the total at the till?

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

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

How many zeros are there at the end of the number which is the product of first hundred positive integers?

Is there an efficient way to work out how many factors a large number has?

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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

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?

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. . . .

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.

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.

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

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

Explore the relationship between simple linear functions and their graphs.

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

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

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

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

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.

The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.

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?

Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.

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 examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

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?

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.

A game that tests your understanding of remainders.

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