Problem 52 In the following exercises, find... [FREE SOLUTION]

Chapter 1: Problem 52

In the following exercises, find the prime factorization. \(455 \)

Short Answer

Expert verified

The prime factorization of 455 is \( 5 \times 7 \times 13 \).

Step by step solution

Divide by the smallest prime number

Start by dividing 455 by the smallest prime number, which is 2. Since 455 is an odd number, it cannot be divided by 2. Move to the next smallest prime number, 3. The sum of the digits of 455 is 4 + 5 + 5 = 14, which is not divisible by 3. Next, try 5. Since 455 ends in 5, it is divisible by 5. So, divide 455 by 5: \(\frac{455}{5} = 91\)

Divide 91 by the smallest suitable prime number

Now, take 91 and check its divisibility by the smallest prime numbers. It isn鈥檛 divisible by 2, 3, or 5, so try 7. Since \(\frac{91}{7} = 13\), and 13 is a prime number, the prime factorization process is complete.

Combine the prime factors

Write out the prime factors obtained in the previous steps. The prime factorization of 455 is: \( 455 = 5 \times 7 \times 13 \)

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Key Concepts

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

Divisibility Rules

Divisibility rules are handy shortcuts that help you determine if one number is divisible by another without using long division. These rules are often the first step in prime factorization. For instance, here鈥檚 how you can use the rules:

Divisibility by 2: A number is divisible by 2 if it is even. Since 455 is odd, it is not divisible by 2.
Divisibility by 3: Add the digits of the number (4 + 5 + 5 = 14). If the sum is divisible by 3, then the original number is too. Since 14 isn't divisible by 3, 455 isn't either.
Divisibility by 5: A number is divisible by 5 if it ends in 0 or 5. Since 455 ends in 5, it is divisible by 5.

These simple checks save time and make factorization easier. The more you practice, the faster you'll get at using these rules.

Prime Numbers

Prime numbers are the building blocks of all numbers. A prime number has only two divisors: 1 and itself. This means it cannot be divided evenly by any other number. Here are some properties:

First few primes: 2, 3, 5, 7, 11, 13, 17, and so on.
Unique factorization: Any number can be uniquely expressed as a product of primes, known as its prime factorization. This is what we do with 455, breaking it down into 5, 7, and 13.
Importance: Prime numbers are crucial in various fields such as cryptography, coding theory, and number theory.

Knowing prime numbers can help you quickly check divisibility and perform factorization efficiently.

Factorization Steps

Factorizing a number means breaking it down into its prime components. For the number 455, you can follow these steps:

Step 1: Start with the smallest prime: Check if the number is divisible by primes starting from 2. Since 455 is odd, it鈥檚 not divisible by 2. The next smallest prime, 3, also doesn鈥檛 work because the sum of the digits is 14, which is not divisible by 3. The prime 5 works because 455 ends in 5. Dividing, we get 455 梅 5 = 91.
Step 2: Proceed with the quotient: Now focus on 91. It鈥檚 not divisible by 2, 3, or 5, but it is by 7. Dividing, we get 91 梅 7 = 13. Since 13 is a prime number, we are done.
Step 3: Write the final product: Combine the prime factors: 5, 7, and 13. Therefore, 455 = 5 脳 7 脳 13.

Following these steps ensures you don't miss any prime factors, and you correctly break down the number.

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91影视

In the following exercises, find the prime factorization. \(455 \)

Short Answer

Step by step solution

Divide by the smallest prime number

Divide 91 by the smallest suitable prime number

Combine the prime factors

Key Concepts

Divisibility Rules

Prime Numbers

Factorization Steps

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