Problem 29 - Elementary and Intermediate Algebra

Question

Find the prime factorization of each number. If the number is prime, state this. $$ 43 $$

Step-by-Step Solution

Answer

43 is a prime number.

Step 1: Identify if the Number is Prime

To determine if 43 is a prime number, check if it has any divisors other than 1 and itself.

Step 2: Check Divisibility by Common Prime Numbers

Check if 43 is divisible by the smallest prime numbers (2, 3, 5, 7, 11, etc.). If 43 is not divisible by any of these, then it is a prime number.

Step 3: Conclude the Prime Factorization

Since 43 is not divisible by any of the smaller prime numbers, it is a prime number. Hence, the prime factorization of 43 is simply 43.

Key Concepts

prime numbers

A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. In other words, a prime number has exactly two distinct positive divisors: 1 and itself. Common examples of prime numbers include 2, 3, 5, and 7.

Prime numbers are like the building blocks of all natural numbers. This is because every number can be expressed as a product of prime numbers (this is called prime factorization). For example, the number 12 can be written as the product of 2, 2, and 3.

In the given exercise, the number 43 is checked to see if it is a prime number. It was determined that 43 has no divisors other than 1 and 43 itself, making it a prime number.

divisibility rules

Divisibility rules are shortcuts that help you figure out if one number can be divided by another without actually performing the division. These rules are particularly useful when you need to check if a number is prime.

Some common divisibility rules include:

• If a number is even, it is divisible by 2.
• If the sum of a number's digits is divisible by 3, then the number is also divisible by 3.
• If a number ends in 0 or 5, it is divisible by 5.

For the given number, 43, checking divisibility by these common primes can quickly show that 43 is not divisible by any of them. This helps in confirming that 43 is a prime number without having to do more extensive calculations.

It’s important to carefully apply these rules. If none of these basic divisibility rules apply to a number, it may very well be a prime.

number theory

Number theory is a branch of mathematics devoted to the study of the integers and more specifically, integer properties and relationships. Key topics within number theory include properties such as prime numbers, divisibility, and the greatest common divisor.

One of the important concepts in number theory is the Fundamental Theorem of Arithmetic, which states that every integer greater than 1 can be represented uniquely as a product of prime factors, up to the order of the factors.

When you perform prime factorization, you are applying this theorem by breaking down a number into its fundamental prime components. In the exercise involving the number 43, number theory principles are used to test the divisibility and confirm that 43 is indeed a prime number. Thus, its prime factorization is simply 43.

By understanding and applying the principles of number theory, we can gain deeper insights into the nature of numbers and their interrelationships. This makes mathematical tasks like factorization not only easier but also more systematic and reliable.