Identifying Prime Numbers from a Given List

The question asks us to determine how many of the given numbers are prime. The numbers provided are 437, 797, and 1073.

What is a Prime Number?

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Numbers that are not prime and are greater than 1 are called composite numbers.

How to Check if a Number is Prime

To check if a number 'n' is prime, we can test if it is divisible by any prime number from 2 up to the square root of 'n'. If 'n' is not divisible by any of these primes, then 'n' is a prime number.

Let's apply this method to each of the given numbers:

Checking the Primality of 437

• First, we find the square root of 437.
• $\sqrt{437} \approx 20.9$.
• We need to check for divisibility by prime numbers up to 19 (since 19 is the largest prime less than 20.9). The prime numbers to check are 2, 3, 5, 7, 11, 13, 17, 19.
• 437 is not divisible by 2, 3, 5, 7, 11, 13, or 17 (you can check this by performing division).
• Let's try dividing 437 by 19:
• $437 \div 19 = 23$.
• Since 437 is divisible by 19 (and 23), it has factors other than 1 and itself.
• Therefore, 437 is a composite number, not a prime number.

Checking the Primality of 797

• Next, we find the square root of 797.
• $\sqrt{797} \approx 28.2$.
• We need to check for divisibility by prime numbers up to 23 (since 23 is the largest prime less than 28.2). The prime numbers to check are 2, 3, 5, 7, 11, 13, 17, 19, 23.
• Upon checking, 797 is not divisible by any of these prime numbers.
• Since 797 is not divisible by any prime number less than or equal to its square root, it has no positive divisors other than 1 and 797.
• Therefore, 797 is a prime number.

Checking the Primality of 1073

• Finally, we find the square root of 1073.
• $\sqrt{1073} \approx 32.7$.
• We need to check for divisibility by prime numbers up to 31 (since 31 is the largest prime less than 32.7). The prime numbers to check are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31.
• 1073 is not divisible by 2, 3, 5, 7, 11, 13, 17, 19, or 23.
• Let's try dividing 1073 by 29:
• $1073 \div 29 = 37$.
• Since 1073 is divisible by 29 (and 37), it has factors other than 1 and itself.
• Therefore, 1073 is a composite number, not a prime number.

Summary of Results

Let's summarize our findings in a table:

From the table, we can see that only one of the given numbers (797) is a prime number.

Therefore, the correct answer is that only one of the numbers is prime.

Revision Table: Understanding Prime Numbers

Additional Information: Large Prime Number Tests

For very large numbers, checking divisibility by all primes up to the square root becomes computationally expensive. There are more advanced primality tests for large numbers, such as:

• Miller-Rabin test (a probabilistic test)
• AKS primality test (a deterministic test)

These tests are used in cryptography and other fields that rely on the properties of large prime numbers. However, for numbers like 437, 797, and 1073, the simple square root method is effective.