Question

Use this definition: A prime number is a positive whole number with no factors other than itself and $1 .$ For example, $2,13,$ and 37 are primes, but 24 and 39 are not. $B y$ convention 1 is not considered prime, so the list of the first few primes is as follows: $2,3,5,7,11,13,17,19,23,29, \ldots$ Let $f$ be the function that assigns to each natural number $x$ the number of primes that are less than or equal to $x .$ For example, $f(12)=5$ because, as you can easily check, five primes are less than or equal to $12 .$ Similarly, $f(3)=2$ because two primes are less than or equal to $3 .$ Find $f(8)$ $f(10),$ and $f(50)$

Name: Problem 63, Chapter 3: Precalculus
Uploaded: 2020-07-26T17:35:55-08:00
Duration: 52 s
Channel: Kristen Karbon

   Use this definition: A prime number is a positive whole number with no factors other than itself and $1 .$ For example, $2,13,$ and 37 are primes, but 24 and 39 are not. $B y$ convention 1 is not considered prime, so the list of the first few primes is as follows:
$2,3,5,7,11,13,17,19,23,29, \ldots$
Let $f$ be the function that assigns to each natural number $x$ the number of primes that are less than or equal to $x .$ For example, $f(12)=5$ because, as you can easily check, five primes are less than or equal to $12 .$ Similarly, $f(3)=2$ because two primes are less than or equal to $3 .$ Find $f(8)$ $f(10),$ and $f(50)$

Precalculus

David Cohen,… 4th Edition

Chapter 3, Problem 63

Instant Answer

Solved by Expert Kristen Karbon

07/26/2020

Step 1

Let's start with $f(8)$. The prime numbers less than or equal to 8 are 2, 3, 5, and 7. So, there are 4 prime numbers less than or equal to 8. Show more…

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Use this definition: A prime number is a positive whole number with no factors other than itself and $1 .$ For example, $2,13,$ and 37 are primes, but 24 and 39 are not. $B y$ convention 1 is not considered prime, so the list of the first few primes is as follows: $2,3,5,7,11,13,17,19,23,29, \ldots$ Let $f$ be the function that assigns to each natural number $x$ the number of primes that are less than or equal to $x .$ For example, $f(12)=5$ because, as you can easily check, five primes are less than or equal to $12 .$ Similarly, $f(3)=2$ because two primes are less than or equal to $3 .$ Find $f(8)$ $f(10),$ and $f(50)$

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Key Concepts

Prime Numbers

In number theory, a prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself. Primes are essential because they serve as the basic building blocks for all natural numbers, following the fundamental theorem of arithmetic, which states that every integer greater than 1 can be uniquely factored into prime numbers.

Prime Counting Function

The prime counting function is a mathematical function that, for a given natural number, returns the number of prime numbers less than or equal to that number. This function helps in understanding the distribution of prime numbers among the integers and is central to many results in analytic number theory, including the study of asymptotic behaviors via the Prime Number Theorem.

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