To find the prime factors of 4567, we need to determine which prime numbers multiply together to give 4567. We can do this by performing prime factorization.

Here’s the step-by-step process:

1. **Check for divisibility by 2**: 4567 is odd, so it is not divisible by 2.
2. **Check for divisibility by 3**: The sum of the digits of 4567 is \(4 + 5 + 6 + 7 = 22\), which is not divisible by 3, so 4567 is not divisible by 3.
3. **Check for divisibility by 5**: 4567 does not end in 0 or 5, so it is not divisible by 5.
4. **Check for divisibility by 7**: \(4567 \div 7 \approx 652.4286\), which is not an integer, so 4567 is not divisible by 7.
5. **Check for divisibility by 11**: \(4 - 5 + 6 - 7 = -2\), which is not divisible by 11, so 4567 is not divisible by 11.
6. **Check for divisibility by 13**: \(4567 \div 13 \approx 351.3077\), which is not an integer, so 4567 is not divisible by 13.
7. **Check for divisibility by 17**: \(4567 \div 17 \approx 268.6471\), which is not an integer, so 4567 is not divisible by 17.
8. **Check for divisibility by 19**: \(4567 \div 19 = 240.3684\), which is not an integer, so 4567 is not divisible by 19.
9. **Check for divisibility by 23**: \(4567 \div 23 \approx 198.5652\), which is not an integer, so 4567 is not divisible by 23.
10. **Check for divisibility by 29**: \(4567 \div 29 \approx 157.4828\), which is not an integer, so 4567 is not divisible by 29.
11. **Check for divisibility by 31**: \(4567 \div 31 \approx 147.3226\), which is not an integer, so 4567 is not divisible by 31.
12. **Check for divisibility by 37**: \(4567 \div 37 = 123.4324\), which is not an integer, so 4567 is not divisible by 37.
13. **Check for divisibility by 41**: \(4567 \div 41 \approx 111.3902\), which is not an integer, so 4567 is not divisible by 41.
14. **Check for divisibility by 43**: \(4567 \div 43 \approx 106.2326\), which is not an integer, so 4567 is not divisible by 43.
15. **Check for divisibility by 47**: \(4567 \div 47 \approx 97.1702\), which is not an integer, so 4567 is not divisible by 47.
16. **Check for divisibility by 53**: \(4567 \div 53 = 86.0\), which is an integer, so 4567 is divisible by 53.

Dividing 4567 by 53:
\[ 4567 \div 53 = 86 \]

Next, we factor 86:
1. **Check for divisibility by 2**: 86 is even, so it is divisible by 2.
\[ 86 \div 2 = 43 \]

Now, we factor 43:
1. **Check for divisibility by 2**: 43 is odd, so it is not divisible by 2.
2. **Check for divisibility by 3**: \(4 + 3 = 7\), which is not divisible by 3, so 43 is not divisible by 3.
3. **Check for divisibility by 5**: 43 does not end in 0 or 5, so it is not divisible by 5.
4. **Check for divisibility by 7**: \(43 \div 7 \approx 6.1429\), which is not an integer, so 43 is not divisible by 7.

43 is a prime number.

Therefore, the prime factorization of 4567 is:
\[ 4567 = 53 \times 2 \times 43 \]

Would you like details or have any questions about this topic?

Here are 8 related questions you might find interesting:
1. How do you find the greatest common divisor (GCD) of two numbers using prime factorization?
2. Can you explain the difference between prime numbers and composite numbers?
3. How do you find the least common multiple (LCM) of two numbers using prime factorization?
4. What is the Euclidean algorithm for finding the GCD of two numbers?
5. Can you explain how to perform the prime factorization of a large number efficiently?
6. What are some common methods for testing the primality of a number?
7. How do you find the prime factors of a number using a factor tree?
8. Can you explain the role of prime numbers in cryptography?

**Tip:** To quickly check the divisibility of a number by small primes, use divisibility rules for numbers like 2, 3, 5, and 11.

Explore the prime factors of 4567 with a detailed step-by-step prime factorization process. Understand how to determine which prime numbers multiply together to give 4567. Suitable for advanced high school students and anyone interested in number theory and prime factorization.

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