Problem 117 Create factor trees for each num... [FREE SOLUTION]

91影视

Prealgebra

David Arnold

$Math Studyset 91影视 Explanations$ Math

2020 Edition

Chapter 1: Problem 117

Create factor trees for each number. Write the prime factorization for each number in compact form, using exponents. $$105$$

Short Answer

Expert verified

The prime factorization of 105 is $3 \times 5 \times 7$.

Step by step solution

Begin Prime Factorization

Write the number 105 and start finding prime factors by dividing it by the smallest prime number, which is 2. Since 105 is odd, proceed to the next prime number, which is 3.

Divide by 3

Check if 105 is divisible by 3. Since the sum of the digits $1 + 0 + 5 = 6$ is divisible by 3, 105 is also divisible by 3. Divide 105 by 3 to get $105 \div 3 = 35$. Record 3 as a prime factor.

Move to Next Prime Number

Now find prime factors for 35. Check the smallest prime factor of 35 which is not divisible by 3 but the next is 5. Divide 35 by 5 to get $35 \div 5 = 7$. Record 5 as another prime factor.

Check for Remaining Prime Number

We are left with 7, which is a prime number. So, 7 is the last prime factor.

Write Prime Factorization in Compact Form

Now that we have all prime factors of 105, write them in compact form using exponents. Since there are no repeated prime factors, the prime factorization is $3^1 \times 5^1 \times 7^1$.

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Key Concepts

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

Factor Trees

A factor tree is a simple tool to help visually break down a number into its prime factors. To create a factor tree:

Start with the original number, in this case, 105.
Choose any two numbers whose product is 105, for instance, 3 and 35.
Write 3 and 35 below 105, and continue breaking down 35 since it is not a prime number.
Split 35 into 5 and 7 and write them below it.
Since both 5 and 7 are prime numbers, you can stop factoring at this point.

Factor trees can vary slightly depending on which factors you choose first. However, the list of prime factors remains the same. Using a factor tree not only assists in finding a number's prime factors but also makes the process easy to visualize and understand. This visualization helps ensure that you don't miss any prime factors, especially for those just getting started with factoring.

Prime Numbers

Prime numbers are the fundamental building blocks in mathematics, especially in prime factorization. A prime number is a whole number greater than 1 that cannot be exactly divided by any number other than 1 or itself.

Examples include numbers like 2, 3, 5, 7, 11, and so forth.
They do not have divisors other than 1 and the number itself.
The smallest prime number is 2, and it is the only even prime number.

Understanding prime numbers is crucial when factorizing a number, as the goal is to break the number down to these indivisible components. Knowing the sequence of prime numbers can assist in efficiently executing prime factorization, such as when processing larger numbers.

Exponents

Exponents provide a more compact way to write repeated multiplication of the same factors. In the context of prime factorization, they help express the result concisely.

An exponent tells you how many times to multiply the base number by itself.
For example, instead of writing 3 multiplied by itself four times as $3 \times 3 \times 3 \times 3$, you can use the notation $3^4$.
In our prime factorization of 105, each prime number is only used once, so you see $3^1$, $5^1$, and $7^1$ in the expression.

Writing the prime factorization in terms of exponents, especially when factors repeat, ensures clarity and simplicity, helping to easily read and manage even complex expressions.

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91影视

Create factor trees for each number. Write the prime factorization for each number in compact form, using exponents. $$105$$

Short Answer

Step by step solution

Begin Prime Factorization

Divide by 3

Move to Next Prime Number

Check for Remaining Prime Number

Write Prime Factorization in Compact Form

Key Concepts

Factor Trees

Prime Numbers

Exponents

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