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Chapter 1: Problem 163

Find the prime factorization of the number. $$83$$

Short Answer

Expert verified
The prime factorization of 83 is 83.

Step by step solution

01

Recognize the Number

Firstly, recognize that the given number is 83.
02

Identifying the Properties of the Number

Determine if 83 is a prime number or not. A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers, meaning it has no positive divisors other than 1 and itself. This is true for 83, so it is indeed a prime number.
03

Prime Factorization of the Number

The prime factorization of a prime number is the number itself. So, the prime factorization of 83 is \(83\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Prime Numbers
Prime numbers are like the building blocks of all natural numbers. A prime number is a number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

Here's what makes a number prime:
  • It has exactly two distinct positive divisors: 1 and itself.
  • This means no other numbers can divide it without leaving a remainder.
For example, numbers such as 2, 3, 5, and 83 are prime because they meet these criteria.

The number 83, specifically, has no divisors other than 1 and 83 itself, confirming its prime status. When we say a number is prime, we imply its simplicity and indivisibility by any other natural number.
Exploring Natural Numbers
Natural numbers are the set of positive integers starting from 1 and extending indefinitely. They form the most basic set of numbers used in counting and ordering.

Natural numbers include:
  • 1, 2, 3, 4,... and continue without end.
These numbers do not include zero, negative numbers, or fractions.

Understanding natural numbers is essential in mathematics as they form the basis from which other number systems and mathematical concepts are built. Since prime numbers are a subset of natural numbers, recognizing this connection helps in identifying whether a given number, like 83, fits the criteria of being prime.
Diving into Divisors
Divisors, or factors, are numbers that evenly divide a given number without leaving a remainder. Knowing about divisors can help determine whether a number is prime.

A prime number has only two divisors:
  • The number 1
  • The number itself
For instance, with the number 83, only 1 and 83 divide it perfectly, confirming it has no other divisors.

In contrast, a composite number has more than two divisors. Understanding divisors helps us easily perform tasks like prime factorization and figure out the underlying structure of a number.

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