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Chapter 3: Problem 70

Simplify. Find the prime factorization of \(150 .[2.1 \mathrm{~d}]\)

Short Answer

Expert verified
The prime factorization of 150 is 2 脳 3 脳 5 脳 5.

Step by step solution

01

Identify the smallest prime factor

The smallest prime factor of 150 is 2. Check if 150 is divisible by 2. Since 150 is even, it is divisible by 2. Factor out 2 from 150: 150 梅 2 = 75.
02

Identify the next smallest prime factor

Next, take 75 and identify its smallest prime factor. 75 is not even, so it is not divisible by 2. Check for divisibility by 3. The sum of the digits in 75 (7 + 5 = 12) is divisible by 3. Therefore, 75 is also divisible by 3. Factor out 3 from 75: 75 梅 3 = 25.
03

Continue with prime factorization

Next, take 25 and identify its smallest prime factor. 25 is not divisible by 2 or 3 but is divisible by 5. Factor out 5 from 25: 25 梅 5 = 5.
04

Complete the factorization

Lastly, take the remaining 5 and factor it out: 5 梅 5 = 1. The prime factorization of 150 is the product of all the factors we found: 150 = 2 脳 3 脳 5 脳 5.

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Key Concepts

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

Simplifying Numbers
Simplifying numbers is a way to break down a complex number into simpler parts. When we simplify, we often use the prime factorization method. Prime factorization involves breaking a number down into its smallest components: prime numbers. For the number 150, we followed a step-by-step process to find its prime factors. Simplifying complex numbers makes it easier to work with them in various mathematical operations. Breaking down numbers to their prime factors can help you understand their properties better, such as divisibility and relationships with other numbers.
Divisibility Rules
Divisibility rules are quick ways to check if a number can be divided by another number without leaving a remainder. These rules are very useful in prime factorization. Here are some important divisibility rules we used:
  • Rule for 2: A number is divisible by 2 if it is even. For example, 150 is even, so it's divisible by 2.
  • Rule for 3: A number is divisible by 3 if the sum of its digits is divisible by 3. For example, the digits of 75 add up to 12, and 12 is divisible by 3.
  • Rule for 5: A number is divisible by 5 if it ends in 0 or 5. For example, 25 ends in 5.
Using these rules can significantly speed up the factorization process and make it more accurate.
Prime Numbers
Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. They are the building blocks of all natural numbers because any number can be expressed as a product of prime numbers. Here are some examples of prime numbers: 2, 3, 5, 7, 11, etc.
  • Why are they important? Prime numbers are crucial for number theory and many areas of mathematics. They are used in algorithms, cryptography, and error-checking.
  • Prime Factorization: To find the prime factors of a number, you start by dividing it by the smallest prime number (2), and continue dividing the quotient by the smallest prime numbers until you reach 1.
In our example, the prime factorization of 150 is achieved through the primes 2, 3, and 5, leading to
150 = 2 脳 3 脳 5 脳 5.

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