Answered

What is the prime factorization of 42?


A$2 \times 3 \times 4$
B$3 \times 3 \times 4$
C$3 \times 3 \times 7$
D$2 \times 3 \times 7$

Verified Answer :

  • Define prime factorization as finding the set of prime numbers whose product equals the target number.
  • Evaluate each option: calculate the products and confirm the primality of the factors.
  • Identify that \(2 \times 3 \times 7 = 42\) and all factors are prime.
  • The correct choice is \(\boxed{d}\).

Explanation

  1. Understanding Prime Factorization

To determine the prime factorization of the number \(42\), we look for a set of prime numbers that, when multiplied together, result in \(42\). A prime number is a natural number greater than \(1\) that has no positive divisors other than \(1\) and itself. We need to check which of the provided options correctly represents this product and consists entirely of prime numbers.

  1. Evaluating the Given Options

We test each option provided to identify the correct prime factorization of \(42\):

  • Option (a): \(2 \times 3 \times 4 = 24\). Since the product is not \(42\), this option is incorrect. Furthermore, \(4\) is not a prime number.
  • Option (b): \(3 \times 3 \times 4 = 36\). Since the product is not \(42\), this option is incorrect.
  • Option (c): \(3 \times 3 \times 7 = 63\). Since the product is not \(42\), this option is incorrect.
  • Option (d): \(2 \times 3 \times 7 = 42\). Here, the product is \(42\), and the numbers \(2\), \(3\), and \(7\) are all prime. This is the correct factorization.
  1. Conclusion

Based on our evaluation, the only expression that correctly factorizes \(42\) into its prime components is \(2 \times 3 \times 7\). Therefore, the correct option is (d). The final answer is \(\boxed{d}\).

Examples

Prime factorization is a fundamental building block in mathematics with widespread applications, particularly in computer security and cryptography. For instance, public-key encryption systems, like RSA, rely on the difficulty of factorizing very large numbers into their prime constituents. Understanding how to break down smaller numbers like \(42 = 2 \times 3 \times 7\) provides the foundation for these complex calculations used to secure sensitive data on the internet.