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Chapter 2: Problem 426

In the following exercises, find the prime factorization of each number. $$ 165 $$

Short Answer

Expert verified
The prime factorization of 165 is \( 3 \times 5 \times 11 \).

Step by step solution

01

Find the smallest prime factor

Begin by finding the smallest prime number that divides 165. Check 2, 3, 5, etc. Since 165 is not even, it is not divisible by 2. Add the digits (1 + 6 + 5 = 12), which is divisible by 3. Therefore, 165 is divisible by 3.
02

Divide by the smallest prime factor

Divide 165 by 3 to find the quotient: \( 165 \div 3 = 55 \)
03

Find the smallest prime factor of the quotient

Next, find the smallest prime factor of the quotient, 55. Since 55 ends in 5, it is divisible by 5.
04

Divide by the smallest prime factor again

Divide 55 by 5 to find the quotient: \( 55 \div 5 = 11 \)
05

Check the remaining quotient

Finally, check if the quotient, 11, is a prime number. Since 11 is only divisible by 1 and itself, it is a prime number.
06

Compile the prime factors

The prime factors of 165 are 3, 5, and 11. Therefore, the prime factorization is \( 165 = 3 \times 5 \times 11 \)

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Key Concepts

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

smallest prime factor
To perform prime factorization, we need to start with the smallest prime factor of a number. Prime factors are prime numbers that multiply together to give the original number. Begin by checking if the number is divisible by the smallest prime numbers. The smallest prime number is 2, followed by 3, 5, and so on. Each time you find a factor, divide the number and proceed with the quotient.
Finding the smallest prime factor manually could involve:
  • Checking divisibility by 2 for even numbers
  • Using specific rules for 3, 5, and other primes
divisibility rules
Divisibility rules are shortcuts to determine if one number divides another without performing division. These rules help identify prime factors more quickly.
Here are some key divisibility rules:
  • A number is divisible by 2 if it is even.
  • A number is divisible by 3 if the sum of its digits is divisible by 3.
  • A number is divisible by 5 if it ends in 0 or 5.
  • A number is divisible by 11 if the difference between the sum of its digits in odd positions and the sum in even positions is divisible by 11.
Using these rules simplifies the process of breaking down numbers into prime factors, enabling faster and easier factorization.
prime numbers
Prime numbers are the building blocks of all numbers. A prime number is a natural number greater than 1, with no divisors other than 1 and itself. Examples include 2, 3, 5, 7, 11, and 13. These numbers have exactly two distinct positive divisors.
Understanding prime numbers is crucial because,
  • They help in factorizing larger numbers
  • They form the foundation for number theory and various algorithms
  • They ensure uniqueness in prime factorization, as every number can be expressed uniquely as a product of primes (Fundamental Theorem of Arithmetic)
Knowing prime numbers and their properties allows for accurate and efficient factorization of any number.

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