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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 collection of resources to support work on Factors and Multiples at Secondary level.
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 number...
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
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?
Have you seen this way of doing multiplication ?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Explore the relationship between simple linear functions and their graphs.
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?
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 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 power.
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 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. . . .
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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?
You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.
This article takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.
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. . . .
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?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
Given the products of diagonally opposite cells - can you complete this Sudoku?
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.
A game that tests your understanding of remainders.
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?
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 hit?
A game in which players take it in turns to choose a number. Can you block your opponent?
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.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
Find the frequency distribution for ordinary English, and use it to help you crack the code.
Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?
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.
Substitution and Transposition all in one! How fiendish can these codes get?
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?
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?
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?
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...
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?
Can you work out what size grid you need to read our secret message?
Follow this recipe for sieving numbers and see what interesting patterns emerge.
Make a line of green and a line of yellow rods so that the lines differ in length by one (a white rod)
Which pairs of cogs let the coloured tooth touch every tooth on the other cog? Which pairs do not let this happen? Why?
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. . . .
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.
The clues for this Sudoku are the product of the numbers in adjacent squares.
What is the smallest number of answers you need to reveal in order to work out the missing headers?
Find the highest power of 11 that will divide into 1000! exactly.