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. . . .
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 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.
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 three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Can you find any perfect numbers? Read this article to find out more...
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.
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
Can you explain the strategy for winning this game with any target?
Is there an efficient way to work out how many factors a large number has?
What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?
Which pairs of cogs let the coloured tooth touch every tooth on the other cog? Which pairs do not let this happen? Why?
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
How many zeros are there at the end of the number which is the product of first hundred positive integers?
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?
Given the products of adjacent cells, can you complete this Sudoku?
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?
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?
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?
The five digit number A679B, in base ten, is divisible by 72. What are the values of A and B?
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?
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.
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. . . .
Follow this recipe for sieving numbers and see what interesting patterns emerge.
Can you work out what size grid you need to read our secret message?
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
Substitution and Transposition all in one! How fiendish can these codes get?
Can you find a way to identify times tables after they have been shifted up?
You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.
Find the highest power of 11 that will divide into 1000! exactly.
How many integers between 1 and 1200 are NOT multiples of any of the numbers 2, 3 or 5?
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?
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.
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. . . .
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
Given the products of diagonally opposite cells - can you complete this Sudoku?
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?"
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.
In how many ways can the number 1 000 000 be expressed as the product of three positive integers?
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?
What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A