Chapter 3: Problem 19
Which of the following are prime numbers? 16, 17, 31, 53, 79
Short Answer
Expert verified
17, 31, 53, and 79 are prime numbers.
Step by step solution
01
Understand the Definition of Prime Numbers
A prime number is any natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A prime number has exactly two distinct positive divisors: 1 and itself.
02
Test for 16
Check if 16 is divisible by any number other than 1 and 16. 16 can be divided by 2, 4, and 8. Therefore, 16 is not a prime number.
03
Test for 17
Check if 17 is divisible by any number other than 1 and 17. 17 cannot be divided evenly by any other number. Therefore, 17 is a prime number.
04
Test for 31
Check if 31 is divisible by any number other than 1 and 31. 31 cannot be divided evenly by any other number. Therefore, 31 is a prime number.
05
Test for 53
Check if 53 is divisible by any number other than 1 and 53. 53 cannot be divided evenly by any other number. Therefore, 53 is a prime number.
06
Test for 79
Check if 79 is divisible by any number other than 1 and 79. 79 cannot be divided evenly by any other number. Therefore, 79 is a prime number.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Number Theory
Number theory is the branch of mathematics that deals with the properties and relationships of numbers. It's focused on the study of natural numbers and integers. Primarily, number theory involves exploring intriguing questions about numbers, such as their divisors, their distribution, and the concept of primes.
Prime numbers are a fundamental topic in number theory. They act as the 'building blocks' of all natural numbers. Much of number theory revolves around understanding and working with prime numbers. For example, the concept of prime factorization, where every natural number greater than 1 can be expressed as a product of prime numbers, originates here.
Learning and mastering prime numbers lays a strong foundation for further studies in number theory and other areas of mathematics.
Prime numbers are a fundamental topic in number theory. They act as the 'building blocks' of all natural numbers. Much of number theory revolves around understanding and working with prime numbers. For example, the concept of prime factorization, where every natural number greater than 1 can be expressed as a product of prime numbers, originates here.
Learning and mastering prime numbers lays a strong foundation for further studies in number theory and other areas of mathematics.
Divisibility
Divisibility refers to whether one number can be divided by another without leaving a remainder. To check for the divisibility of a number, identify its factors. Factors are numbers that can be multiplied together to get the original number.
In the case of prime numbers, a number is prime if it has no divisors other than 1 and itself. For a number to not be prime, it must be divisible by at least one other number.
For example, when we test 16 for being a prime number, we see that it can be divided by 2, 4, and 8, proving that it has divisors other than 1 and itself. Hence, 16 is not a prime number.
In contrast, when we test 17, we find no divisors other than 1 and 17 itself, proving 17 is a prime number.
In the case of prime numbers, a number is prime if it has no divisors other than 1 and itself. For a number to not be prime, it must be divisible by at least one other number.
For example, when we test 16 for being a prime number, we see that it can be divided by 2, 4, and 8, proving that it has divisors other than 1 and itself. Hence, 16 is not a prime number.
In contrast, when we test 17, we find no divisors other than 1 and 17 itself, proving 17 is a prime number.
Natural Numbers
Natural numbers are the set of positive integers starting from 1, 2, 3, and so on. They are the most basic and straightforward type of numbers used in counting and ordering.
Prime numbers fall under the category of natural numbers. Understanding natural numbers is crucial as they form the base of more complex arithmetic and number theory concepts.
Natural numbers also include composite numbers, which are the numbers that have more than two distinct positive divisors.
In summary, the natural numbers include both prime and composite numbers, each having its significance in number theory and divisibility.
Prime numbers fall under the category of natural numbers. Understanding natural numbers is crucial as they form the base of more complex arithmetic and number theory concepts.
Natural numbers also include composite numbers, which are the numbers that have more than two distinct positive divisors.
In summary, the natural numbers include both prime and composite numbers, each having its significance in number theory and divisibility.