Problem 26 - Elementary and Intermediate Algebra

Question

Find the prime factorization of each number. If the number is prime, state this. $$ 98 $$

Step-by-Step Solution

Answer

98 = 2 × 7²

Step 1: Check If 98 is a Prime Number

First, determine if 98 is a prime number by checking if it has any divisors other than 1 and itself. A prime number has no divisors other than these two.

Step 2: Find Initial Prime Factor

Since 98 is even, it is divisible by 2. Divide 98 by 2: To simplify: to divide 98 by 2 to get 49: Divide 98 by 2 to get 49: \[ \frac{98}{2} = 49 \]

Step 3: Further Factorize

Next, factorize 49. Notice that 49 is 7 times 7 or, mathematically: \[ 49 = 7 \times 7 \] This indicates that 7 is a prime number, and 49 is the square of 7.

Step 4: Write the Prime Factorization

Combine the factors found in the previous steps to write the full prime factorization of 98. Thus, the prime factorization of 98 is: \[ 98 = 2 \times 7^2 \]

Key Concepts

factors

In mathematics, factors are numbers you can multiply together to get another number. For example, the factors of 10 are 1, 2, 5, and 10, because:

• 1 × 10 = 10
• 2 × 5 = 10

When we talk about 'prime factorization', we mean expressing a number as a product of its prime factors. A prime factor is a factor that is a prime number, such as 2, 3, or 7. In the example provided, we find that the factors of 98 include both 2 and 7.
To find the prime factorization of a number, follow these steps:

• Start with the smallest prime (2) and divide step-by-step.
• If the number isn't divisible by a smaller prime, move to the next larger one.
• Continue until the result itself is a prime number.

This way, each factor you write out in the process is a prime factor.

divisibility

Divisibility is an essential concept when working with factors and prime factorization. A number is divisible by another if, after division, there is no remainder. For instance, 98 is divisible by 2 because:

• When 98 is divided by 2, the result is 49, and there is no remainder.

We say that 2 is a divisor of 98.
To check for divisibility, you can follow these rules for the smallest prime numbers:

• A number is divisible by 2 if it is even (ends in 0, 2, 4, 6, or 8).
• 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.

After you've checked divisibility by smaller numbers, continue with larger primes until you've fully factorized your number.

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. For example, 2, 3, 5, 7, 11, and 13 are all prime numbers.
When performing prime factorization, you're breaking down a number into its prime components. In the given example, 98 is broken down into 2 and 7, because they are the primes that multiply together to make 98:

• First, divide by 2 (smallest prime number), giving you 49.
• Then, realize that 49 is 7 times 7; and 7 is a prime number.

This means 98 = 2 × 7 × 7 or 2 × 7². Recognizing and understanding prime numbers makes it easier to follow the steps of prime factorization and understand why these steps are taken.