Problem 8
Question
In the following exercises, find the prime factorization. 78
Step-by-Step Solution
Verified Answer
Prime factorization of 78 is \[ 2 × 3 × 13 \].
1Step 1 – Divide by the smallest prime number
Divide 78 by the smallest prime number, which is 2: \[ 78 ÷ 2 = 39 \]
2Step 2 – Continue with the next smallest prime number
Next, divide 39 by the smallest prime number that is a factor of 39, which is 3: \[ 39 ÷ 3 = 13 \]
3Step 3 – Identify prime factor
Check if 13 is a prime number. Since 13 is prime, it cannot be factored further.
4Step 4 – Write the prime factorization
Combine the prime factors found in the previous steps: \[ 78 = 2 × 3 × 13 \]
Key Concepts
Divisibility RulesPrime NumbersFactor Trees
Divisibility Rules
To find prime factors, understanding divisibility rules is very helpful. These rules help to quickly determine if a number can be divided by another without performing lengthy division.
For instance, to check if a number is divisible by 2, it must be even. In other words, its last digit should be 0, 2, 4, 6, or 8.
For 3, if the sum of the digits of the number is divisible by 3, then the number itself is divisible by 3.
For 5, the last digit must be 0 or 5.
This helps avoid unnecessary trial and error when searching for factors, speeding up the process of prime factorization.
For instance, to check if a number is divisible by 2, it must be even. In other words, its last digit should be 0, 2, 4, 6, or 8.
For 3, if the sum of the digits of the number is divisible by 3, then the number itself is divisible by 3.
For 5, the last digit must be 0 or 5.
This helps avoid unnecessary trial and error when searching for factors, speeding up the process of prime factorization.
Prime Numbers
Prime numbers are the building blocks of all numbers. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.
Common examples include 2, 3, 5, 7, 11, 13, and so on.
When factoring a number, we break it down into prime numbers.
For example, in the exercise, we identified 2, 3, and 13 as prime numbers while factoring 78.
Prime numbers play a critical role in various branches of mathematics, especially number theory.
Common examples include 2, 3, 5, 7, 11, 13, and so on.
When factoring a number, we break it down into prime numbers.
For example, in the exercise, we identified 2, 3, and 13 as prime numbers while factoring 78.
Prime numbers play a critical role in various branches of mathematics, especially number theory.
Factor Trees
A factor tree is a helpful tool to visualize the factorization process. It breaks down a number step-by-step into prime factors.
Start with the number you need to factor, and divide it by the smallest prime number.
Continue this process until all remaining factors are prime numbers.
The end goal is to outline a 'tree' where the roots are the original numbers, the branches are intermediate steps, and the leaves are the prime factors.
Factor trees visually help trace back the prime components of the number.
Start with the number you need to factor, and divide it by the smallest prime number.
- In this example, we started with 78. Divide by 2 to get 39.
- Then, divide 39 by 3 to get 13.
Continue this process until all remaining factors are prime numbers.
The end goal is to outline a 'tree' where the roots are the original numbers, the branches are intermediate steps, and the leaves are the prime factors.
Factor trees visually help trace back the prime components of the number.
Other exercises in this chapter
Problem 6
In the following exercises, use the divisibility tests to determine whether each number is divisible by 2 , by \(3,\) by \(5,\) by \(6,\) and by 10 . 39,075
View solution Problem 7
In the following exercises, find the prime factorization. 86
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In the following exercises, find the prime factorization. 455
View solution Problem 11
In the following exercises, find the prime factorization. 432
View solution