Problem 8 Find two different factorization... [FREE SOLUTION]

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Prealgebra

Julie Miller, Molly O鈥橬eill, Nancy Hyde

$Math Studyset 91影视 Explanations$ Math

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Chapter 4: Problem 8

Find two different factorizations of each number. (Answers may vary.) $$20$$

Short Answer

Expert verified

Two factorizations: 20 = 2 脳 10 or 20 = 4 脳 5.

Step by step solution

Prime Factorization

Find the prime factorization of 20. This means breaking down the number into prime factors.Start by noting that 20 is an even number, so it is divisible by 2:\[ 20 \ 2 \times 10 \]Next, factor 10:\[ 10 \ 2 \times 5 \]Putting it all together, we get:\[ 20 = 2 \times 2 \times 5 = 2^2 \times 5 \]

First Factorization

Use the prime factors to create the first factorization. Group the factors in two different ways:First way:\[ 20 = 2 \times 10 \]

Second Factorization

Create a second factorization by grouping the prime factors differently:Second way:\[ 20 = 4 \times 5 \ \left( since \ 4 = 2^2 \right) \]

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Key Concepts

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

Factorization

Factorization is the process of breaking down a number into smaller numbers that multiply together to get the original number.
In this context, we focus on breaking numbers down into prime factors.
The prime factors are the building blocks that, when multiplied together, give back the original number.
For example, the prime factorization of 20 is done by dividing it by the smallest prime number possible until we can't divide anymore:

20 is divisible by 2 (since 20 is even): 20 梅 2 = 10

Next, 10 is also divisible by 2: 10 梅 2 = 5

5 is a prime number. Thus, the prime factorization of 20 is 2 脳 2 脳 5, which we can also write as 2² 脳 5.

These factors can then be grouped to create different factorizations like 2 脳 10 or 4 脳 5.

Prime Numbers

Prime numbers are numbers greater than 1 that have only two factors - 1 and themselves.
They cannot be divided evenly by any other number.
Prime numbers play a crucial role in factorization because they are the 'atoms' of our number system - the indivisible elements from which all numbers are constructed.
Some examples of prime numbers are:

When we look at the number 20, it is not a prime number because it can be divided by numbers other than 1 and 20.
However, through the process of factorization, we break it down into 2, 2, and 5, which are all prime numbers.
This makes prime numbers the key elements in achieving the complete factorization of any composite number.

Multiplication

Multiplication is the mathematical operation where we add a number to itself a certain number of times.
In factorization, we use multiplication to put our prime factors back together.
We check if the product equals the original number.
For instance, in our factorization of 20:

Prime factors are 2, 2, and 5:
2 脳 2 脳 5
If we multiply these, $2 脳 2 = 4$, which results in $ 4 脳 5 = 20 $.

Another way to multiply for 20:
We can combine the prime factors differently.
This results in: $ 4 (since 2^2 = 4) 脳 5 = 20 $

Different ways of grouping these factors can create different multiplication groups:

2 脳 10 = 20
4 脳 5 = 20

Understanding how multiplication ties into factorization helps in verifying the correctness of our factorized form.

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91影视

Find two different factorizations of each number. (Answers may vary.) $$20$$

Short Answer

Step by step solution

Prime Factorization

First Factorization

Second Factorization

Key Concepts

Factorization

Prime Numbers

Multiplication

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