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.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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...
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...
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?
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...
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?
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.
I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
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. . . .
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
Is there an efficient way to work out how many factors a large number has?
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.
Ben, Jack and Emma passed counters to each other and ended with the same number of counters. How many did they start with?
How many noughts are at the end of these giant numbers?
Find the number which has 8 divisors, such that the product of the divisors is 331776.
Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.
Can you find any two-digit numbers that satisfy all of these statements?
Can you explain the strategy for winning this game with any target?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Got It game for an adult and child. How can you play so that you know you will always win?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Play this game and see if you can figure out the computer's chosen number.
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
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...
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?
Can you work out what size grid you need to read our secret message?
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
Can you find a way to identify times tables after they have been shifted up or down?
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.
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
Each letter represents a different positive digit AHHAAH / JOKE = HA What are the values of each of the letters?
Find the highest power of 11 that will divide into 1000! exactly.
Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?
Can you work out how many lengths I swim each day?
The clues for this Sudoku are the product of the numbers in adjacent squares.
The items in the shopping basket add and multiply to give the same amount. What could their prices be?
You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.
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. . . .
I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?
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?
What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?
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. . . .
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
A game in which players take it in turns to choose a number. Can you block your opponent?