What is the prime factorization of $50 ?$
A. $50=5 \times 5 \times 2$
B. $50=2 \times 2 \times 5$
C. $50=10 \times 5$
D. $50=2 \times 2 \times 2 \times 5
Answer (2)
We want to find the prime factorization of 50.
Divide 50 by the smallest prime number, 2, to get 25.
Divide 25 by the smallest prime number that divides it, 5, to get 5.
Express 50 as the product of its prime factors: 50 = 2 × 5 × 5 .
Explanation
Problem Analysis We are asked to find the prime factorization of the number 50. This means we want to express 50 as a product of prime numbers. A prime number is a number greater than 1 that has only two factors: 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, and so on.
Finding the First Prime Factor To find the prime factors of 50, we can start by dividing 50 by the smallest prime number, which is 2. 50 ÷ 2 = 25 . So, 2 is a prime factor of 50. Now we need to find the prime factors of 25.
Finding the Next Prime Factor The smallest prime number that divides 25 is 5. 25 ÷ 5 = 5 . So, 5 is a prime factor of 25. Now we need to find the prime factors of 5.
Complete Prime Factorization Since 5 is a prime number, we have found all the prime factors of 50. The prime factors are 2, 5, and 5. Therefore, the prime factorization of 50 is 50 = 2 × 5 × 5 .
Final Answer Thus, the prime factorization of 50 is 50 = 2 × 5 × 5 .
Examples
Prime factorization is a fundamental concept in number theory with various real-world applications. For instance, in cryptography, prime factorization is used to ensure the security of encrypted data. Imagine you want to send a secure message over the internet. You can encrypt the message using a large number that is the product of two prime numbers. If someone tries to intercept the message, they would need to factorize this large number to decode it. The larger the prime numbers, the harder it is to factorize, making the message more secure. This is a simplified example, but it illustrates how prime factorization plays a crucial role in modern encryption techniques.
The prime factorization of 50 is given by the expression 50 = 2 × 5 × 5 or equivalently, 50 = 2 × 5 2 . This means 2 and 5 are the prime factors of 50, with 5 appearing twice. The correct answer from the options provided is A. 50 = 5 × 5 × 2 .
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