ERATOSTHENES SIEVE
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Eratosthenes, a Greek mathematician from the 3rd century BC, developed a simple sieve method for identifying prime numbers. The method works by sieving out multiples of successive numbers, leaving the prime numbers behind. It involves making a list of consecutive numbers up to a certain number, crossing out numbers that are multiples of 2, then crossing out multiples of 3, and so on, leaving the remaining uncrossed numbers as the prime numbers. The document then proceeds to explain Eratosthenes' sieve method in more detail through 5 steps.
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A prime number is a whole number greater than 1 that is only divisible by 1 and itself. A composite number is one that has divisors other than 1 and itself. The document then explains that prime numbers follow a pattern of being multiples of 6 plus or minus 1 (6n±1), and provides a two-step process for determining if a number is prime or composite: 1) check if it is a multiple of 6 plus or minus 1, and 2) check if it is divisible by any numbers up to its square root.
Division, divisibility rules 11 3-10
This document outlines divisibility rules to determine factors of numbers based on their digits. The rules are: if the ones digit is even, the number is divisible by 2; if the sum of the digits is divisible by 3, the number is divisible by 3; if the number ends in 0 or 5, it is divisible by 5; if the sum of the digits is divisible by 9, the number is divisible by 9; and if a number ends in 0, it is divisible by 10. Examples are provided for each rule.
Prime Numbers
This document discusses prime numbers, composite numbers, and how to determine which category a whole number falls into. It provides that:
- Prime numbers have exactly two unique factors - the number itself and 1.
- Composite numbers have more than two factors.
- Only the numbers 0, 1, and 2 are neither prime nor composite - 0 because every number divides it, and 1 because it only has one factor. All other whole numbers are either prime or composite.
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ERATOSTHENES SIEVE
- 1. The Eratosthenes' Sieve Back in 200 BC a Greek mathematician named Eratosthenes developed a simple step-by-step method for determining which numbers are prime. This method is called "Eratosthenes' Sieve".
- 2. A sieve is a strainer that allows small particles to pass through while catching the larger particles.
- 3. In the case of Eratosthenes' Sieve, the small particles that pass through are all the non-prime numbers and the large particles that remain are the Prime Numbers.
- 4. Eratosthenes
- 5. He was born in Cirene (actually Shahhat, Libia, which is in Africa) He studied in Alexandria Arquimedes was his friend He died at the age of 80.
- 6. He was a great mathematician, philosopher, poet and astronomer. People knew him as the second Plato or because he was always second in every subject he studied.
- 7. His method to find out the prime numbers can be divided into 5 simple steps.
- 8. But wait a minute, folks!
- 9. What are Prime Numbers? Numbers Divisible Only by Themselves and 1 Any number that has only two factors, itself and 1.
- 10. Is the number 1 a prime number? No, it isn´t. The number 1 is NOT considered Prime Why? Because it only has one factor , itself.
- 11. What is a sieve? What can we do with Eratosthenes´ sieve? We can find out which of the numbers from 1 to 100 are prime Let´s do it!
- 12. The Eratosthenes' Sieve - Step 1 Make a table of the numbers 1 to 100 organized into 10 rows
- 13. The Eratosthenes' Sieve - Step 2 Cross out the number 1 because by definition it is NOT Prime number
- 14. The Eratosthenes' Sieve - Step 3 Circle the number 2 and cross out all the multiples of 2 (or all the even numbers)
- 15. The Eratosthenes' Sieve - Step 4 Look for the following number that is not crossed out. Circle the number and cross out all the multiples of that number.
- 16. The Eratosthenes' Sieve - Step 5 Repeat step 4 until you get to the number 7.
- 18. Now you can watch a simulation of the sieve...Pay attention!