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?
Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?
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.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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?
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?
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.
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
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
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?
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
What is the smallest number with exactly 14 divisors?
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. . . .
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. . . .
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
You are given the Lowest Common Multiples of sets of digits. Find
the digits and then solve the Sudoku.
Have you seen this way of doing multiplication ?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the. . . .
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). . . .
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
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.
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
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.
The clues for this Sudoku are the product of the numbers in adjacent squares.
Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
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?
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 puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.
Can you find a relationship between the number of dots on the
circle and the number of steps that will ensure that all points are
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?
Can you find any perfect numbers? Read this article to find out more...
The five digit number A679B, in base ten, is divisible by 72. What
are the values of A and B?
Given the products of adjacent cells, can you complete this Sudoku?
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.
Find the highest power of 11 that will divide into 1000! exactly.
Can you find a way to identify times tables after they have been shifted up?
Each letter represents a different positive digit
AHHAAH / JOKE = HA
What are the values of each of the letters?
Explore the relationship between simple linear functions and their
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...
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?"
Some 4 digit numbers can be written as the product of a 3 digit
number and a 2 digit number using the digits 1 to 9 each once and
only once. The number 4396 can be written as just such a product.
Can. . . .
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
A collection of resources to support work on Factors and Multiples at Secondary level.
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?
Find the number which has 8 divisors, such that the product of the
divisors is 331776.