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Chapter 2: Problem 272

In the following exercises, find the prime factorization of each number using the factor tree method. $$ 420 $$

Short Answer

Expert verified
420 = 2^2 脳 3 脳 5 脳 7

Step by step solution

01

- Start with the given number

Write the number 420 at the top of your factor tree.
02

- Find two factors of 420

Choose any two numbers that multiply to give 420. For example, 2 and 210 (since 420 = 2 脳 210). Draw branches down from 420 and write 2 and 210.
03

- Factor the remaining composite number (210)

Next, take the number 210 and find two factors of it. For instance, 2 and 105 (since 210 = 2 脳 105). Draw branches down from 210 and write 2 and 105.
04

- Continue factoring the composite numbers

Now, take the number 105 and find its factors. For example, 3 and 35 (since 105 = 3 脳 35). Draw branches down from 105 and write 3 and 35.
05

- Factor 35 further

Finally, take the number 35 and factor it into 5 and 7 (since 35 = 5 脳 7). Draw branches down from 35 and write 5 and 7.
06

- List all prime factors

Now, list out all the prime numbers at the ends of the branches: 2, 2, 3, 5, and 7.
07

- Write the prime factorization

Combine the prime factors to express 420 as a product of primes: 420 = 2^2 脳 3 脳 5 脳 7.

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Key Concepts

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

Factor Tree Method
The factor tree method is a simple way to find the prime factors of a composite number. Imagine a tree where the given number is the root, and branches grow down as we break the number into smaller factors. To start:
  • Write the number at the top.
  • Then, find two factors of that number.
  • If these factors are not prime, keep factoring them until only prime numbers remain.
This visually shows how a number can be broken down to its prime components and makes complex numbers easier to handle.
Composite Numbers
Composite numbers are numbers that have more than two distinct positive divisors: 1 and the number itself, plus at least one additional divisor. In simpler terms, a composite number can be divided evenly by numbers other than 1 and itself. For example:
  • 4 is composite because it can be divided by 1, 2, and 4.
  • 15 is composite because it can be divided by 1, 3, 5, and 15.
Recognizing composite numbers is essential when using the factor tree method since they can be broken down further into smaller factors.
Prime Numbers
Prime numbers are the opposite of composite numbers. A prime number only has two distinct positive divisors: 1 and itself. In other terms, these numbers cannot be evenly divided by any other number. Examples include:
  • 2, as it can only be divided by 1 and 2.
  • 3, as it can only be divided by 1 and 3.
Prime numbers are the building blocks of all numbers. Finding prime numbers is the ultimate goal of the factor tree method. Once you've broken a number down to primes, you're done.
Multiplication
Understanding multiplication is crucial for both creating and checking factor trees. Here's a quick recap:
  • Multiplication is combining groups of equal sizes. For example, 4 脳 3 means 4 groups of 3.
  • In the factor tree method, you're essentially reversing multiplication by breaking numbers down to their smaller factors.
  • To verify the prime factorization, you multiply the prime factors together to get the original number. For instance, with 420: 2^2 脳 3 脳 5 脳 7 should equal 420.
Mastering multiplication helps in both constructing and double-checking your work with factor trees.

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