Problem 21 - Introductory and Intermediate Algebra for College Students 4th

Question

Identify each natural number as prime or composite. If the number is composite, find its prime factorization. $$140$$

Step-by-Step Solution

Answer

The number 140 is a composite number. The prime factorization of 140 is: \(2 * 2 * 5 * 7\) or equivalently \(2^2 * 5 * 7\).

Step 1: Identify the Number Type

The given number is 140. As it has more than two distinct divisors (it's divisible by 1, 2, 5, 7, etc.), it's deemed a composite number.

Step 2: Start the Prime Factorization

Start to factorize by dividing 140 by the smallest prime number, which is 2. The result is 70.

Step 3: Continue the Prime Factorization

Two is still a divisor of 70, so divide 70 by 2 again. The result is 35.

Step 4: Further Prime Factorization

Now the smallest prime number that can divide 35 is 5, so divide 35 by 5. The result is 7.

Step 5: Finalize Prime Factorization

Lastly, 7 is a prime number itself, so the process stops here.

Key Concepts

Composite Number

A composite number is a fascinating concept in mathematics. It is a natural number that has more than two distinct divisors. This means that, unlike a prime number, a composite number can be divided evenly by numbers other than 1 and itself.
For example, the number 140 is composite because it can be divided by the divisors: 1, 2, 5, 7, 10, 14, 20, 28, 35, 70, and 140.

• A quick way to recognize a composite number is to check if it is divisible by any prime number less than its square root.
• Composite numbers always have a prime factorization, which is the product of prime numbers that equals the composite number itself.

Understanding the nature of composite numbers helps in many areas of mathematics, such as simplifying fractions and finding least common multiples.

Prime Number

A prime number is essentially the opposite of a composite number. It is a natural number greater than 1, which can only be divided by 1 and itself without leaving a remainder.
This means that a prime number has exactly two distinct divisors.

• Examples of prime numbers are 2, 3, 5, 7, and 11.
• Importantly, the number 2 is the smallest and the only even prime number.

Prime numbers are the building blocks of all natural numbers since any number can be expressed as a product of prime numbers. This is known as prime factorization.

Natural Numbers

Natural numbers are the set of positive integers beginning from 1 and continuing infinitely (i.e., 1, 2, 3, ...).
These are the numbers we use for counting and ordering in everyday situations.

• Natural numbers do not include zero or negative numbers.
• Prime and composite numbers are subsets of natural numbers.

Understanding natural numbers is foundational because they form the basis of arithmetic, including addition, subtraction, multiplication, and division.

Divisors

A divisor is a number that divides another number completely without leaving any remainder.
Divisors are critical for determining if a number is prime or composite.

• In the case of 140, the divisors include 1, 2, 5, 7, and so on.
• Finding all divisors of a number can help perform operations such as finding the greatest common divisor (GCD) or simplifying fractions.

Divisors play a crucial role in our understanding of numbers and their properties. Knowing how to find divisors helps in computations involving factorization and number theory.