Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
A collection of resources to support work on Factors and Multiples at Secondary level.
Can you find a way to identify times tables after they have been shifted up?
A game that tests your understanding of remainders.
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.
Given the products of diagonally opposite cells - can you complete this Sudoku?
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.
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
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?
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
What is the remainder when 2^2002 is divided by 7? What happens
with different powers of 2?
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?
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?
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?
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.
A game in which players take it in turns to choose a number. Can you block your opponent?
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
For this challenge, you'll need to play Got It! Can you explain the
strategy for winning this game with any target?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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?
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. . . .
Find the frequency distribution for ordinary English, and use it to help you crack the code.
Substitution and Transposition all in one! How fiendish can these codes get?
Can you work out what size grid you need to read our secret message?
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?
Given the products of adjacent cells, can you complete this Sudoku?
How many zeros are there at the end of the number which is the
product of first hundred positive integers?
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...
Explore the factors of the numbers which are written as 10101 in
different number bases. Prove that the numbers 10201, 11011 and
10101 are composite in any base.
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.
Make a line of green and a line of yellow rods so that the lines
differ in length by one (a white rod)
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. . . .
Each letter represents a different positive digit
AHHAAH / JOKE = HA
What are the values of each of the letters?
You are given the Lowest Common Multiples of sets of digits. Find
the digits and then solve the Sudoku.
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
The clues for this Sudoku are the product of the numbers in adjacent squares.
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). . . .
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?"
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
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?
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. . . .
In how many ways can the number 1 000 000 be expressed as the
product of three positive integers?
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
This article takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.
This package contains a collection of problems from the NRICH
website that could be suitable for students who have a good
understanding of Factors and Multiples and who feel ready to take
on some. . . .
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?
Do you know a quick way to check if a number is a multiple of two?
How about three, four or six?