What is the prime factorization of [tex]$\underline{28}$[/tex]?
A. [tex]$28=2 \times 14$[/tex]
B. [tex]$28=2 \times 2 \times 3 \times 3$[/tex]
C. [tex]$28=2 \times 2 \times 7$[/tex]
D. [tex]$28=4 \times 7$[/tex]
Answer (2)
The problem asks for the prime factorization of 28.
Divide 28 by the smallest prime number, 2, to get 28 = 2 × 14 .
Divide 14 by 2 to get 14 = 2 × 7 .
The prime factorization of 28 is therefore 2 × 2 × 7 .
Explanation
Understanding Prime Factorization We are asked to find the prime factorization of 28. This means we need to express 28 as a product of prime numbers. A prime number is a number greater than 1 that has only two factors: 1 and itself.
First Step: Dividing by 2 We can start by dividing 28 by the smallest prime number, which is 2. 28 ÷ 2 = 14 So, we can write 28 = 2 × 14 .
Second Step: Dividing 14 by 2 Now we need to factor 14. We can again divide by the smallest prime number, 2. 14 ÷ 2 = 7 So, we can write 14 = 2 × 7 .
Combining the Factors Now we have 28 = 2 × 14 = 2 × ( 2 × 7 ) = 2 × 2 × 7 . Since 2 and 7 are both prime numbers, we have found the prime factorization of 28.
Final Answer Therefore, the prime factorization of 28 is 2 × 2 × 7 . Comparing this to the given options, we see that option C is the correct answer.
Examples
Prime factorization is a fundamental concept in number theory and has many practical applications. For example, when designing secure communication systems, prime factorization is used to create encryption keys. The security of these systems relies on the fact that it is computationally difficult to factor very large numbers into their prime factors. In everyday life, understanding prime factors can help in simplifying fractions or finding the greatest common divisor of two numbers, which can be useful in various scenarios like dividing items into equal groups.
The prime factorization of 28 is expressed as 2 × 2 × 7 . By dividing successively by the smallest prime number, we arrive at this expression. Therefore, the correct answer is option C.
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