Problem 25 Find the prime factorization of ... [FREE SOLUTION]

Chapter 1: Problem 25

Find the prime factorization of each number. If the number is prime, state this. $$ 27 $$

Short Answer

Expert verified

27 = 3^3.

Step by step solution

Check If 27 is a Prime Number

Determine if 27 is a prime number by checking if it has only two distinct positive divisors: 1 and itself. Since 27 = 3 脳 9, 27 is not a prime number.

Start Dividing by the Smallest Prime Number

Begin the factorization process by dividing 27 by the smallest prime number, which is 3. ewline 27 梅 3 = 9

Continue Dividing 9

Next, factor 9, since it is not a prime number either. ewline 9 梅 3 = 3

Divide 3 Again

Finally, factor 3 since it is a prime number. ewline 3 梅 3 = 1

List All Prime Factors

Now list all the prime factors obtained from each step: ewline 27 = 3 脳 3 脳 3 or 27 = 3^3.

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Key Concepts

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

Prime Numbers

Prime numbers are special numbers. A prime number is a number greater than 1 with only two positive divisors: 1 and itself. This means that a prime number cannot be divided evenly by any other number.
For example, 2, 3, 5, 7, 11, and 13 are all prime numbers. They have no other divisors apart from 1 and themselves.
Identifying a prime number is the first step in many factorization problems. If a number can't be divided by any prime less than or equal to its square root, then it is a prime number.

Divisors

Divisors are numbers that divide another number completely without leaving a remainder. For instance, the divisors of 6 are 1, 2, 3, and 6. They divide 6 without any remainder.
To find the prime factorization of a number, you start by finding its smallest divisors. In the example exercise, we first identified that 27 wasn鈥檛 prime because it could be divided by 3, one of its divisors. By continually finding and dividing by the smallest prime divisors, we break down the original number into prime factors.

Multiplication

Multiplication is one of the basic arithmetic operations used frequently in factorization problems. After identifying all prime factors of a number, we use multiplication to represent the original number.
In the exercise given, we broke down 27 as follows: 27 divided by 3 equals 9, 9 divided by 3 equals 3, and 3 divided by 3 equals 1. So, 27 can be written as 3 脳 3 脳 3. This can also be expressed using powers of a number, where 27 = 3^3.
Understanding multiplication is vital in putting back together the prime factors to their original number.

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91影视

Find the prime factorization of each number. If the number is prime, state this. $$ 27 $$

Short Answer

Step by step solution

Check If 27 is a Prime Number

Start Dividing by the Smallest Prime Number

Continue Dividing 9

Divide 3 Again

List All Prime Factors

Key Concepts

Prime Numbers

Divisors

Multiplication

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