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One method for finding prime numbers is the sieve of Eratosthenes. The natural numbers from 2 to 50 are shown in the table. Start at the number 2 (the smallest prime number). Leave the number 2 and cross out every second number after the number $2 .$ This will eliminate all numbers that are multiples of $2 .$ Then go back to the beginning of the chart and leave the number $3,$ but cross out every third number after the number 3 (thus eliminating the multiples of 3 ). Begin at the next open number and continue this process. The numbers that remain are prime numbers. Use this process to find the prime numbers less than $50 .$ $$ \begin{array}{r|r|r|r|r|r|r|r|r|r} & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\ \hline 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 \\ \hline 31 & 32 & 33 & 34 & 35 & 36 & 37 & 38 & 39 & 40 \\ \hline 41 & 42 & 43 & 44 & 45 & 46 & 47 & 48 & 49 & 50 \\ \hline \end{array} $$

   One method for finding prime numbers is the sieve of Eratosthenes. The natural numbers from 2 to 50 are shown in the table. Start at the number 2 (the smallest prime number). Leave the number 2 and cross out every second number after the number $2 .$ This will eliminate all numbers that are multiples of $2 .$ Then go back to the beginning of the chart and leave the number $3,$ but cross out every third number after the number 3 (thus eliminating the multiples of 3 ). Begin at the next open number and continue this process. The numbers that remain are prime numbers. Use this process to find the prime numbers less than $50 .$
$$
\begin{array}{r|r|r|r|r|r|r|r|r|r} 
& 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\
\hline 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 \\
\hline 31 & 32 & 33 & 34 & 35 & 36 & 37 & 38 & 39 & 40 \\
\hline 41 & 42 & 43 & 44 & 45 & 46 & 47 & 48 & 49 & 50 \\
\hline
\end{array}
$$

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Prealgebra & Introductory Algebra

Prealgebra & Introductory Algebra

Julie Miller, Molly… 2nd Edition

Chapter 4, Problem 35

Instant Answer

Solved by Expert Victor Salazar

04/08/2022

Step 1

We leave the number 2 and cross out every second number after the number 2. This will eliminate all numbers that are multiples of 2. The numbers that are crossed out are: 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50. Show more…

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One method for finding prime numbers is the sieve of Eratosthenes. The natural numbers from 2 to 50 are shown in the table. Start at the number 2 (the smallest prime number). Leave the number 2 and cross out every second number after the number $2 .$ This will eliminate all numbers that are multiples of $2 .$ Then go back to the beginning of the chart and leave the number $3,$ but cross out every third number after the number 3 (thus eliminating the multiples of 3 ). Begin at the next open number and continue this process. The numbers that remain are prime numbers. Use this process to find the prime numbers less than $50 .$ $$ \begin{array}{r|r|r|r|r|r|r|r|r|r} & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\ \hline 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 \\ \hline 31 & 32 & 33 & 34 & 35 & 36 & 37 & 38 & 39 & 40 \\ \hline 41 & 42 & 43 & 44 & 45 & 46 & 47 & 48 & 49 & 50 \\ \hline \end{array} $$

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Key Concepts

Prime Numbers

Prime numbers are natural numbers greater than 1 that have no divisors other than 1 and themselves. They form the building blocks of arithmetic and are fundamental in various fields of mathematics, including number theory and cryptography, because every number can be uniquely factored into primes.

Sieve of Eratosthenes

The Sieve of Eratosthenes is an ancient algorithm used to find all prime numbers up to a given limit. It systematically eliminates the composite numbers by iteratively crossing out the multiples of each prime starting from 2. The numbers that remain unmarked after this process are the primes.

Multiples and Elimination Technique

This technique is based on the observation that once a prime number is identified, all of its multiples must be composite. By crossing out the multiples of each prime in a systematic manner, the algorithm efficiently filters out all non-prime numbers without the need for repeated division tests.

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