Finding the Prime Factors of 323

To determine the number of prime factors of 323, we first need to find the prime factorization of the number 323. Prime factorization is the process of breaking down a composite number into its prime number components.

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples of prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, and so on.

Step-by-Step Prime Factorization of 323

We can find the prime factors by testing if 323 is divisible by small prime numbers starting from 2.

• Is 323 divisible by 2? No, because 323 is an odd number.
• Is 323 divisible by 3? To check, we sum the digits: $3 + 2 + 3 = 8$. Since 8 is not divisible by 3, 323 is not divisible by 3.
• Is 323 divisible by 5? No, because the last digit is not 0 or 5.
• Is 323 divisible by 7? Let's divide: $323 \div 7$. $7 \times 4 = 28$, leaving 43. $7 \times 6 = 42$, leaving 1. So, $323 = 7 \times 46 + 1$. Not divisible by 7.
• Is 323 divisible by 11? We can use the alternating sum of digits: $3 - 2 + 3 = 4$. Since 4 is not divisible by 11, 323 is not divisible by 11.
• Is 323 divisible by 13? Let's divide: $323 \div 13$. $13 \times 2 = 26$, leaving 63. $13 \times 4 = 52$, leaving 11. So, $323 = 13 \times 24 + 11$. Not divisible by 13.
• Is 323 divisible by 17? Let's divide: $323 \div 17$. We can estimate: $17 \times 10 = 170$, $17 \times 20 = 340$. So the factor is likely between 10 and 20. Let's try: $17 \times 19$. $17 \times (20 - 1) = 17 \times 20 - 17 \times 1 = 340 - 17 = 323$. Yes, $323 \div 17 = 19$.

So, we found that $323 = 17 \times 19$. Now we need to check if 17 and 19 are prime numbers.

• 17 is a prime number because its only positive divisors are 1 and 17.
• 19 is a prime number because its only positive divisors are 1 and 19.

Therefore, the prime factorization of 323 is $17 \times 19$.

The distinct prime factors of 323 are 17 and 19.

Now, we count the number of these distinct prime factors. There are two prime factors: 17 and 19.

Conclusion on Prime Factors of 323

The number 323 has exactly two distinct prime factors, which are 17 and 19. This matches one of the options provided.

Revision Table: Prime Factor Concepts

Additional Information on Prime Factorization

Prime factorization is a fundamental concept in number theory. It is used in many areas, such as finding the greatest common divisor (GCD) and the least common multiple (LCM) of two or more numbers.

Every composite number has a unique prime factorization, according to the Fundamental Theorem of Arithmetic. This means that no matter how you find the prime factors of a number, you will always end up with the same set of prime numbers (though the order might be different).

When counting the number of prime factors, sometimes the question asks for the number of *distinct* prime factors (as in this case, 17 and 19 are distinct), and sometimes it asks for the total number of prime factors, counting multiplicity (e.g., for 12 = $2^2 \times 3$, the distinct prime factors are 2 and 3 (two distinct factors), but the total prime factors are 2, 2, and 3 (three factors in total, counting multiplicity)). In the context of the options provided, "Two prime factors" refers to the distinct prime factors.