A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Is there an efficient way to work out how many factors a large number has?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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?
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...
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 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.
Can you find any perfect numbers? Read this article to find out more...
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. . . .
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.
How many integers between 1 and 1200 are NOT multiples of any of the numbers 2, 3 or 5?
Nine squares are fitted together to form a rectangle. Can you find its dimensions?
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.
What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?
Can you find a way to identify times tables after they have been shifted up or down?
Which pairs of cogs let the coloured tooth touch every tooth on the other cog? Which pairs do not let this happen? Why?
Explore the relationship between simple linear functions and their graphs.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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.
How many different number families can you find?
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 two-digit numbers that satisfy all of these statements?
This article explains various divisibility rules and why they work. An article to read with pencil and paper handy.
I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?
Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?
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?
Can you select the missing digit(s) to find the largest multiple?
A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .
Can you explain the strategy for winning this game with any target?
Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
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?
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
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...
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?
Can you find different ways of creating paths using these paving slabs?
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?"
The flow chart requires two numbers, M and N. Select several values for M and try to establish what the flow chart does.
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" .
Find the highest power of 11 that will divide into 1000! exactly.
This article for teachers describes how number arrays can be a useful representation for many number concepts.
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?
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?
The clues for this Sudoku are the product of the numbers in adjacent squares.
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.
How many numbers less than 1000 are NOT divisible by either: a) 2 or 5; or b) 2, 5 or 7?