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?

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'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 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?

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?

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!?

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

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

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

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

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?

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

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

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

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" .

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

A game that tests your understanding of remainders.

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

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

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

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.

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. . . .

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

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

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.

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

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

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

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

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?

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?

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?

What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?

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

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?"

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

Explore the relationship between simple linear functions and their graphs.

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

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

Each letter represents a different positive digit AHHAAH / JOKE = HA What are the values of each of the letters?

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

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

Can you explain the strategy for winning this game with any target?

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?

Data is sent in chunks of two different sizes - a yellow chunk has 5 characters and a blue chunk has 9 characters. A data slot of size 31 cannot be exactly filled with a combination of yellow and. . . .

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.