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Chapter 5: Problem 22

Find the prime factorization of each number. $$98$$

Short Answer

Expert verified
The prime factorization of 98 is \( 2 \times 7 \times 7 \) or \( 2 \times (7^2) \).

Step by step solution

01

Start with the smallest prime number

Start by dividing 98 by the smallest prime number, which is 2. Since 98 is an even number, it's divisible by 2. 98 divided by 2 is 49.
02

Continue with next prime number

Now we continue the process with 49. 49 is not divisible by 2. The next smallest prime number is 3, and 49 is not divisible by 3 either. So, we move to the next prime number, which is 5 and 49 is not divisible by this number as well. The next prime number is 7, and 49 is divisible by 7. 49 divided by 7 is 7.
03

Final step

Now we're left with 7. Since 7 is a prime number, this is the final step of our factorization. We can't divide 7 by any other prime number except 1 and 7 itself. Hence, stop here.

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Key Concepts

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

Understanding Prime Numbers
Prime numbers are the building blocks of all whole numbers. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. This means it can't be divided evenly without leaving a remainder by any other number.
  • The smallest prime number is 2, which is also the only even prime number.
  • Examples of prime numbers include 3, 5, 7, 11, and so on.
Prime numbers are crucial when we talk about prime factorization because they represent the simplest building blocks that cannot be broken down further into other factors. Recognizing these numbers is the first step in the process of breaking down composite numbers into their prime factors.
Mastering Divisibility Rules
Divisibility rules help you determine if one number is divisible by another without actually performing the division. These rules are helpful in finding prime factors quickly.
  • Divisibility by 2: A number is divisible by 2 if it is even, meaning its last digit is 0, 2, 4, 6, or 8.
  • Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
  • Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.
For example, in the factorization of 98, we first check the smallest number, 2. Since 98 is even, it's divisible by 2. Checking divisibility early helps save time and ensures accuracy in identifying factors.
Efficient Factorization Techniques
Factorization techniques involve systematically breaking a number down into its prime factors. This is often done using division.
  • Start with the smallest prime number and see if it divides the number without leaving a remainder.
  • If it doesn't, move to the next smallest prime number.
  • Continue this process until the number reduces to 1.
This sequential approach helps break down even the largest numbers. For 98:1. Divide by 2 since 98 is even, resulting in 49.2. Check smaller primes for 49: 3 and 5 do not divide 49.3. Divide by 7, which gives 7.4. Since 7 is prime, the process stops here.Prime factorization of 98 is thus expressed as \(2 \times 7^2\), highlighting its prime building blocks.

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Most popular questions from this chapter

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