What is the prime factorization of [tex]33[/tex]?
A. [tex]33=3 \times 11[/tex]
B. [tex]33=3 \times 3 \times 3[/tex]
C. [tex]33=2 \times 3 \times 3 \times 3[/tex]
D. [tex]33=2 \times 3 \times 5[/tex]
Answer (1)
Prime factorization expresses a number as a product of its prime factors.
Check each option to see if it equals 33 and consists of prime numbers.
Option A, 33 = 3 × 11 , is the correct prime factorization since both 3 and 11 are prime.
The prime factorization of 33 is 3 × 11 .
Explanation
Understanding Prime Factorization We are asked to find the prime factorization of 33 from the given options. Prime factorization means expressing a number as a product of its prime factors. A prime number is a number greater than 1 that has only two factors: 1 and itself.
Checking the Options Let's examine the options: Option A: 33 = 3 × 11 . Both 3 and 11 are prime numbers. So, this is a valid prime factorization. Option B: 33 = 3 × 3 × 3 = 27 . This is incorrect because 3 × 3 × 3 equals 27, not 33. Option C: 33 = 2 × 3 × 3 × 3 = 54 . This is incorrect because 2 × 3 × 3 × 3 equals 54, not 33. Option D: 33 = 2 × 3 × 5 = 30 . This is incorrect because 2 × 3 × 5 equals 30, not 33.
Finding the Correct Factorization Therefore, the correct prime factorization of 33 is 3 × 11 .
Final Answer The prime factorization of 33 is 3 × 11 .
Examples
Prime factorization is used in cryptography to ensure secure communication. For example, the security of the RSA algorithm relies on the fact that it is difficult to factor large numbers into their prime factors. In real life, this is used to protect sensitive information such as credit card numbers and personal data when transmitted over the internet. Factoring a number like 33 into its prime factors is a basic example of the kind of math that underlies these security systems.
