Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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
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 largest integer which divides every member of the
following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.
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. . . .
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
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?
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?
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
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?
Helen made the conjecture that "every multiple of six has more
factors than the two numbers either side of it". Is this conjecture
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 three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
A collection of resources to support work on Factors and Multiples at Secondary level.
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?
You are given the Lowest Common Multiples of sets of digits. Find
the digits and then solve the Sudoku.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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. . . .
Have you seen this way of doing multiplication ?
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 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 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?
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). . . .
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.
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.
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?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
A challenge that requires you to apply your knowledge of the
properties of numbers. Can you fill all the squares on the board?
Find the highest power of 11 that will divide into 1000! exactly.
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.
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
The clues for this Sudoku are the product of the numbers in adjacent squares.
Each letter represents a different positive digit
AHHAAH / JOKE = HA
What are the values of each of the letters?
Find the number which has 8 divisors, such that the product of the
divisors is 331776.
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?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
How many integers between 1 and 1200 are NOT multiples of any of
the numbers 2, 3 or 5?
Can you find any perfect numbers? Read this article to find out more...
Make a line of green and a line of yellow rods so that the lines
differ in length by one (a white rod)
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.
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
What is the smallest number of answers you need to reveal in order
to work out the missing headers?
Follow this recipe for sieving numbers and see what interesting patterns emerge.
Can you work out what size grid you need to read our secret message?
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?
Find the frequency distribution for ordinary English, and use it to help you crack the code.