A game that tests your understanding of remainders.
A collection of resources to support work on Factors and Multiples at Secondary level.
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
The five digit number A679B, in base ten, is divisible by 72. What
are the values of A and B?
How many numbers less than 1000 are NOT divisible by either: a) 2
or 5; or b) 2, 5 or 7?
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
Given the products of adjacent cells, can you complete this Sudoku?
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?
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?"
Find the number which has 8 divisors, such that the product of the
divisors is 331776.
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" .
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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.
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. . . .
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 zeros are there at the end of the number which is the
product of first hundred positive integers?
Given the products of diagonally opposite cells - can you complete this Sudoku?
What is the remainder when 2^2002 is divided by 7? What happens
with different powers of 2?
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. . . .
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?
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.
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
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?
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?
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. . . .
Make a line of green and a line of yellow rods so that the lines
differ in length by one (a white rod)
You are given the Lowest Common Multiples of sets of digits. Find
the digits and then solve the Sudoku.
Can you find a way to identify times tables after they have been shifted up?
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?
The clues for this Sudoku are the product of the numbers in adjacent squares.
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
Can you find any perfect numbers? Read this article to find out more...
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 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.
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.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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). . . .
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.
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
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?
Complete the following expressions so that each one gives a four
digit number as the product of two two digit numbers and uses the
digits 1 to 8 once and only once.
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?
I put eggs into a basket in groups of 7 and noticed that I could
easily have divided them into piles of 2, 3, 4, 5 or 6 and always
have one left over. How many eggs were in the basket?
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?