The Sieve of Eratosthenes

Introduction

In this lesson, you will be doing A LOT of colouring... Take a look at this 10 x 10 grid (either PRINT ME or look at this interactive version).

  1. Starting with the number 2, put a box around it then colour in all its multiples. Click on this video if you need to a reminder of what a multiple is.

    • What patterns would you produce?

  2. Now pick the next smallest number, put a box around it then colour in all its multiples in a different colour.

    • What patterns would you produce?

    • Were there numbers that you coloured in more than once?

    • What are these numbers? (You wont be able to have 2 colours in one box for the interactive version).

    • What can you tell me about these numbers?

  3. Repeat step 2 until there is a good reason to stop.

    • What are the numbers that are left?

    • What can you tell me about these numbers?

  4. Check you answer with this interactive version.

Further Questions and Challenges

  • Which multiples have three different colours? What is the smallest multiple with 4 different colours? What about 5 colours, etc? Is the next number after these always a prime? Why?

  • Explain how you can use Sieve of Eratosthenes to find the HCF of 30 and 45. How about the HCF of 52 and 60?

    • HCF stands for the highest common factor. Look at this video for two ways of find it.

  • What percentage of the first 100 numbers is prime? What about the next 100? Does this change as the numbers get bigger? Why/why not?

    • You may wish to think about the following related problem: I put the numbers 1 to 100 in a bag. I select one randomly. What is the probability it is a prime number?

    • I put the numbers 1 to n in a bag. I select one randomly. What is the probability it is a prime number? What is the smallest possible value of n?

    • Draw a graph of probabilities as n increases. Hint: plot "n" on the x-axis and the corresponding probability as a decimal on the y-axis.

  • What is the largest prime that leads to a new multiple being coloured (i.e. one that hasn’t already been used as a multiple of a smaller prime)? How does this vary for different size grids? (You can do this easily on this interactive version).

  • Now change it to a size 6 grid (PRINT ME or look at this interactive version).

    • Which columns do you find most of the prime numbers?

    • Ignoring the top row in the size 6 grid, the primes only appear in two different columns, which is one-third of the available columns. Can this be beaten? Do any of tables have primes in all of their columns?

Further Practice

Some of the essential skills introduced in this lesson are "multiples" and "prime numbers". Other skills you may also wish to recap/ learn about are "prime factorisation", "HCF" and "LCM". Practise these relevant skills on DrFrostMaths, CorbettMaths, MyiMaths and Eedi.


Transum

Try these self-marked exercises on transum to check your understanding: factor trees and HCF and LCM.

Looking for something fun to play with your friend? Try this "Flabbergasted" game. The winner is the last person standing.



Try the crossumber puzzle below:

Extension

Try this Prime Numbers Jigsaw or this Prime Labyrinth on transum! For something more challenging, try this Scallywags puzzle where you have to arrange the scallywags and scoundrels on the chairs so that the numbers of any two sitting next to each other add up to a prime number.


How Many Factors

Take a look at the following activities which leads onto prime factorisation.

How Many Factors.pdf

Try to come up with a conjecture for each of the columns in the table.

  • For example: “Numbers with exactly 2 factors are prime”.

  • Some of your conjecture might have more than one explanation or idea.


Look at the numbers with exactly 3 factors. What can you say about them?

  • Can your predict the next biggest number with exactly 3 factors?

  • How many numbers less than 100 will have exactly 3 factors?


Look at the numbers with exactly 4 factors.

  • There are two types of number in the list.

  • For each of the two types, what would the next largest number be?


45 can be written as 3 × 5 × 5.

  • How many factors do you think 45 has?

  • Check to see if you are correct.

  • How many other numbers below 100 will have the same number of factors?


30 can be written as 2 × 3 × 5.

  • It has exactly eight factors.

  • Find another number than has exactly eight factors.

  • How many other numbers below 100 will have exactly eight factors?


Only one number has exactly 5 factors and no numbers have exactly 7 factors.

  • Copy and complete this table and use it to find a number great than 30 that does have exactly 7 factors?

Prime Products Investigations

Take a look at these investigations below on prime products:

Prime Products - Student Questions.docx
Prime Products - Task Sheet.docx

And how would use prime products to solve these?

  1. There are two whole numbers between 60 and 70 that multiply to be 4095. What are they?


  1. Three pupils in a secondary school and their headteacher all have birthdays on the same day.

    • The product of the ages of the pupils is 2652.

    • The sum of the ages of the pupils is exactly the age of the headteacher.

    • How old is the headteacher?


3. My friend once asked me how old my 3 children are. this is what i told him:

    • The product of their ages is 36

    • The sum of their ages is the same as his house number

    • My oldest kid is blond

    • How old are my kids?

4. Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is.

  • Great Granddad was born in the year A (where A is the product of 3 prime numbers),

  • He was 20 years old in the year B (where B is the product of a prime number and a square number),

  • He was 80 years old in the year C (where C is the product of two prime numbers) and he celebrated his 100th birthday in the year D (where D is even and the product of 4 prime numbers).

  • When was he born?


Transum

  • Hotel Digital - On Mondays the lift will go to any floor, on Tuesdays it will only go to floors which are multiples of 2, on Wednesdays it will only go to floors which are multiples of 3 and so on. Which would be the best floor of the hotel to stay on if you do not like using the stairs and you plan to stay for a week?

  • Verruca Value - The "Verruca Value" of a word is the number of vowels multiplied by the number on consonants in that word.

    • For example: "Mathematics" is 28 (7 consonants, 4 vowels). How many words can you find with a Verruca Value of 24? 23? etc

  • For more goodies on factors, multiples and prime look at these activities on transum.


Don Steward Goodies on HCF/LCM