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.
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...
15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?
Is there an efficient way to work out how many factors a large number has?
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?
Think of any three-digit number. Repeat the digits. The 6-digit number that you end up with is divisible by 91. Is this a coincidence?
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.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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?
Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?
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.
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. . . .
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Can you explain the strategy for winning this game with any target?
Got It game for an adult and child. How can you play so that you know you will always win?
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
Can you find any perfect numbers? Read this article to find out more...
How many integers between 1 and 1200 are NOT multiples of any of the numbers 2, 3 or 5?
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
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?
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 clue in this Sudoku is the product of the two numbers in adjacent cells.
Find the highest power of 11 that will divide into 1000! exactly.
Given the products of adjacent cells, can you complete this Sudoku?
Are these statements always true, sometimes true or never true?
The five digit number A679B, in base ten, is divisible by 72. What are the values of A and B?
Can you find any two-digit numbers that satisfy all of these statements?
Given the products of diagonally opposite cells - can you complete this Sudoku?
Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.
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.
This article for teachers describes how number arrays can be a useful representation for many number concepts.
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.
Have a go at balancing this equation. Can you find different ways of doing it?
What is the smallest number of answers you need to reveal in order to work out the missing headers?
How many different number families can you find?
Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
Can you work out some different ways to balance this equation?
Explore the relationship between simple linear functions and their graphs.
In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?
Which pairs of cogs let the coloured tooth touch every tooth on the other cog? Which pairs do not let this happen? Why?
Does this 'trick' for calculating multiples of 11 always work? Why or why not?
The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?
Play this game and see if you can figure out the computer's chosen number.
Can you find different ways of creating paths using these paving slabs?
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.
The clues for this Sudoku are the product of the numbers in adjacent squares.
Nine squares are fitted together to form a rectangle. Can you find its dimensions?