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.
Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.
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. . . .
Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.
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...
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?
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?
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.
In how many ways can the number 1 000 000 be expressed as the product of three positive integers?
Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.
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 line of green and a line of yellow rods so that the lines differ in length by one (a white rod)
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. . . .
How many noughts are at the end of these giant numbers?
Can you work out what size grid you need to read our secret message?
Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?
This article takes the reader through divisibility tests and how they work. An article to read with pencil and paper to hand.
How many integers between 1 and 1200 are NOT multiples of any of the numbers 2, 3 or 5?
I added together the first 'n' positive integers and found that my answer was a 3 digit number in which all the digits were the same...
What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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?
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.
Explore the factors of the numbers which are written as 10101 in different number bases. Prove that the numbers 10201, 11011 and 10101 are composite in any base.
Find the highest power of 11 that will divide into 1000! exactly.
A collection of resources to support work on Factors and Multiples at Secondary level.
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. . . .
Given the products of diagonally opposite cells - can you complete this Sudoku?
Is there an efficient way to work out how many factors a large number has?
6! = 6 x 5 x 4 x 3 x 2 x 1. The highest power of 2 that divides exactly into 6! is 4 since (6!) / (2^4 ) = 45. What is the highest power of two that divides exactly into 100!?
The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?
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 explain the strategy for winning this game with any target?
Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.
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.
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.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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.
What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A
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.
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?
How did the the rotation robot make these patterns?
Can you find any perfect numbers? Read this article to find out more...
Got It game for an adult and child. How can you play so that you know you will always win?
Follow this recipe for sieving numbers and see what interesting patterns emerge.
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.