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?

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

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

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

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

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

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

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?

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!?

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.

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

I added together the first 'n' positive integers and found that my answer was a 3 digit number in which all the digits were the same...

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?

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?

I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?

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

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

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

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.

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.

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

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

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

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. . . .

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?

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

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

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

Can you find any two-digit numbers that satisfy all of these statements?

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. . . .

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

I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

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 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.

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

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?"

Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?

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

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

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

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

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

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...

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.

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

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?

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

Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). . . .