How many integers between 1 and 1200 are NOT multiples of any of
the numbers 2, 3 or 5?
Follow this recipe for sieving numbers and see what interesting patterns emerge.
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
Make a line of green and a line of yellow rods so that the lines
differ in length by one (a white rod)
A collection of resources to support work on Factors and Multiples at Secondary level.
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?
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?
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.
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?
Which pairs of cogs let the coloured tooth touch every tooth on the
other cog? Which pairs do not let this happen? Why?
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. . . .
Given the products of diagonally opposite cells - can you complete this Sudoku?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
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.
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 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?
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. . . .
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
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?
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?
How many zeros are there at the end of the number which is the
product of first hundred positive integers?
What is the smallest number with exactly 14 divisors?
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?
Given the products of adjacent cells, can you complete this Sudoku?
Find the number which has 8 divisors, such that the product of the
divisors is 331776.
Find the largest integer which divides every member of the
following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
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.
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. . . .
Have you seen this way of doing multiplication ?
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.
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. . . .
A game that tests your understanding of remainders.
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
Each letter represents a different positive digit
AHHAAH / JOKE = HA
What are the values of each of the letters?
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
Find the highest power of 11 that will divide into 1000! exactly.
What is the smallest number of answers you need to reveal in order
to work out the missing headers?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Can you find any perfect numbers? Read this article to find out more...
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?
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?"
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.