Find the largest integer which divides every member of the
following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
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. . . .
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
Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
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
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
How many noughts are at the end of these giant numbers?
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?
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. . . .
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
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). . . .
Make a line of green and a line of yellow rods so that the lines
differ in length by one (a white rod)
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 any perfect numbers? Read this article to find out more...
How many integers between 1 and 1200 are NOT multiples of any of
the numbers 2, 3 or 5?
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?
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. . . .
A collection of resources to support work on Factors and Multiples at Secondary level.
This article takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.
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.
Can you work out what size grid you need to read our secret message?
Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?
You are given the Lowest Common Multiples of sets of digits. Find
the digits and then solve the Sudoku.
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
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.
Follow this recipe for sieving numbers and see what interesting patterns emerge.
What is the smallest number of answers you need to reveal in order
to work out the missing headers?
Substitution and Transposition all in one! How fiendish can these codes get?
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
Given the products of adjacent cells, can you complete this Sudoku?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
The five digit number A679B, in base ten, is divisible by 72. What
are the values of A and B?
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 clues for this Sudoku are the product of the numbers in adjacent squares.
How many zeros are there at the end of the number which is the
product of first hundred positive integers?
Have you seen this way of doing multiplication ?
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. . . .
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
Each letter represents a different positive digit
AHHAAH / JOKE = HA
What are the values of each of the letters?
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 a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Explore the relationship between simple linear functions and their
Which pairs of cogs let the coloured tooth touch every tooth on the
other cog? Which pairs do not let this happen? 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?
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?
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?
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?
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.
In how many ways can the number 1 000 000 be expressed as the
product of three positive integers?