what is the prime factorization of 47?

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31-03-2025

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The prime factorization of a number is finding the prime numbers that, when multiplied together, equal the original number. Let's explore how to find the prime factorization of 47.

Understanding Prime Numbers

Before diving into the factorization of 47, let's quickly review what prime numbers are. A prime number is a whole number greater than 1 that has only two divisors: 1 and itself. For example, 2, 3, 5, 7, and 11 are all prime numbers.

Finding the Prime Factorization of 47

The process of finding the prime factorization of a number involves systematically dividing it by prime numbers until you're left with only prime numbers as factors. Let's apply this to 47:

1. Check for Divisibility by Small Prime Numbers: We start by checking if 47 is divisible by the smallest prime numbers: 2, 3, 5, 7, etc.

2. Testing Divisibility:

   • 47 is not divisible by 2 (it's not an even number).
   • 47 is not divisible by 3 (the sum of its digits, 4 + 7 = 11, is not divisible by 3).
   • 47 is not divisible by 5 (it doesn't end in 0 or 5).
   • 47 is not divisible by 7 (47 ÷ 7 ≈ 6.71).

3. Continuing the Process: We continue testing divisibility by larger prime numbers. However, we can stop once we reach a number whose square is greater than 47. This is because if 47 had a prime factor larger than the square root of 47 (approximately 6.86), it would also have a prime factor smaller than the square root of 47, and we would have already found it.

4. The Result: Since we've tested divisibility by all prime numbers up to 6 and found no factors, we conclude that 47 itself is a prime number.

Therefore, the prime factorization of 47 is simply 47.

Prime Factorization: Beyond 47

Understanding prime factorization is crucial in various mathematical concepts. This includes simplifying fractions, finding the least common multiple (LCM), and greatest common divisor (GCD) of numbers, and even plays a role in cryptography.

While 47's prime factorization is straightforward, remember that larger numbers may have multiple prime factors. For those, a systematic approach using a factor tree or division by prime numbers is necessary.

Frequently Asked Questions (FAQs)

Q: What is a prime number?

A: A prime number is a whole number greater than 1 that is only divisible by 1 and itself.

Q: How do I find the prime factorization of a number?

A: Start by dividing the number by the smallest prime numbers (2, 3, 5, 7, etc.) until you are left with only prime numbers.

Q: Why is the prime factorization of 47 so simple?

A: 47 is itself a prime number. Therefore, its prime factorization is just the number 47.

Q: Are there any online tools to help with prime factorization?

A: Yes, many online calculators and websites can help you find the prime factorization of any number.

This article provides a clear and concise explanation of the prime factorization of 47, including essential background information and frequently asked questions. The simple nature of 47's prime factorization makes it an ideal example to illustrate the concept.