Follow this recipe for sieving numbers and see what interesting patterns emerge.
How many integers between 1 and 1200 are NOT multiples of any of
the numbers 2, 3 or 5?
Make a line of green and a line of yellow rods so that the lines
differ in length by one (a white rod)
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 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?
Given the products of adjacent 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.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. 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?
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. . . .
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
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.
A collection of resources to support work on Factors and Multiples at Secondary level.
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.
Which pairs of cogs let the coloured tooth touch every tooth on the
other cog? Which pairs do not let this happen? Why?
Given the products of diagonally opposite cells - can you complete this Sudoku?
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?
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?
The sum of the first 'n' natural numbers is a 3 digit number in which all the digits are the same. How many numbers have been summed?
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.
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
The five digit number A679B, in base ten, is divisible by 72. What
are the values of A and B?
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?
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
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?
How many zeros are there at the end of the number which is the
product of first hundred positive integers?
Each letter represents a different positive digit
AHHAAH / JOKE = HA
What are the values of each of the letters?
A game that tests your understanding of remainders.
You are given the Lowest Common Multiples of sets of digits. Find
the digits and then solve the Sudoku.
Twice a week I go swimming and swim the same number of lengths of the pool each time. As I swim, I count the lengths I've done so far, and make it into a fraction of the whole number of lengths I. . . .
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Substitution and Transposition all in one! How fiendish can these codes get?
What is the smallest number of answers you need to reveal in order
to work out the missing headers?
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.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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.
Can you find any perfect numbers? Read this article to find out more...
Can you work out what size grid you need to read our secret message?
Find the highest power of 11 that will divide into 1000! exactly.
Have you seen this way of doing multiplication ?
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). . . .
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
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?"
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve 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.
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
In how many ways can the number 1 000 000 be expressed as the
product of three positive integers?
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. . . .