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Chapter 1: Problem 17

Find the prime factorization of each number. $$ 28 $$

Short Answer

Expert verified
The prime factorization of 28 is 2 脳 2 脳 7.

Step by step solution

01

Understand Prime Factorization

Prime factorization is the process of breaking down a number into its basic prime number building blocks. A prime number is a natural number greater than 1 that is not divisible by any other numbers except 1 and itself.
02

Start with the Smallest Prime

The smallest prime number is 2. Check if 28 is divisible by 2. Since 28 is an even number, it is divisible by 2. Divide 28 by 2: 28 梅 2 = 14.
03

Continue Factoring 14

Now, take 14. Again, check divisibility by 2. Since 14 is also even, divide by 2: 14 梅 2 = 7.
04

Check if 7 is Prime

Now we have 7. Determine if 7 is a prime number. A prime number is only divisible by 1 and itself, and since 7 fits this, it is a prime number.
05

Combine Prime Factors

All the divisors are prime, so the prime factorization of 28 is complete. Combine your results: 28 = 2 脳 2 脳 7.

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Key Concepts

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

Prime Numbers
Prime numbers serve as the fundamental components for prime factorization. These are numbers greater than 1, which have no other divisors apart from 1 and themselves. For instance, 2, 3, 5, and 7 are prime numbers because they cannot be divided evenly by any other numbers.
  • They are key because every natural number greater than 1 has a unique prime factorization.
  • Think of them as the "atoms" of the number world, building blocks for other numbers.
  • When you identify prime numbers, you make it easier to break down more complex numbers.
Recognizing these numbers helps you appreciate their role in mathematics and applications like cryptography.
Factoring
Factoring is the process of breaking down a number into smaller numbers, called factors, that when multiplied together return the original number. For instance, factoring 28 gives us numbers like 2, 4, 7, and 14.
  • The goal of prime factorization is to keep factoring until all factors are prime numbers.
  • Start with the smallest prime number and work upwards.
  • In the example of 28, we divide by 2 until we can no longer do so, and then we test other prime numbers, like 3, 5, or 7.
Factoring helps simplify problems and is essential for solving polynomial equations, among other applications.
Natural Numbers
Natural numbers are the counting numbers that are used in everyday life. They start from 1 and go on infinitely (1, 2, 3, ...). These are distinct from whole numbers because they do not include zero.
  • Prime numbers and factoring primarily consider positive natural numbers.
  • They are used for counting, ordering, and for defining other mathematical operations.
  • Knowing the properties of natural numbers assists in mathematical problem-solving and comprehension of foundational math concepts.
By understanding natural numbers, primality becomes easier to grasp, leading to a stronger foundation in mathematics.

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Most popular questions from this chapter

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