Understanding Unique Prime Divisors

The question asks for the number of unique prime divisors of 960. A divisor of a number is a number that divides it evenly. A prime divisor is a divisor that is also a prime number. Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves (examples: 2, 3, 5, 7, 11, etc.). We are interested in the unique prime numbers that divide 960.

Finding Prime Divisors through Prime Factorization

The most reliable way to find the prime divisors of any number is through prime factorization. Prime factorization is the process of breaking down a composite number into a product of its prime factors. Once we have the prime factorization, we can simply list the distinct prime numbers that appear in the product.

Step-by-Step Prime Factorization of 960

Let's find the prime factorization of 960:

\(960\)

We can start by dividing 960 by the smallest prime number, which is 2, as it is an even number:

\(960 \div 2 = 480\)

480 is also even:

\(480 \div 2 = 240\)

240 is even:

\(240 \div 2 = 120\)

120 is even:

\(120 \div 2 = 60\)

60 is even:

\(60 \div 2 = 30\)

30 is even:

\(30 \div 2 = 15\)

15 is not divisible by 2. The next smallest prime number is 3. 15 is divisible by 3:

\(15 \div 3 = 5\)

5 is a prime number.

So, the prime factorization of 960 is \(2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 5\).

We can write this in exponential form as \(2^6 \times 3^1 \times 5^1\).

Identifying the Unique Prime Divisors

From the prime factorization \(2^6 \times 3^1 \times 5^1\), the prime numbers that divide 960 are the bases of the powers in the factorization. These are 2, 3, and 5.

The unique prime divisors are:

• 2
• 3
• 5

To find the number of unique prime divisors, we simply count how many distinct prime numbers are in this list. There are three unique prime divisors: 2, 3, and 5.

Therefore, the number of unique prime divisors of 960 is 3.

Revision Table: Number Theory Concepts

Additional Information: Prime Numbers and Divisors

Every composite number has a unique prime factorization, according to the Fundamental Theorem of Arithmetic. This theorem states that any integer greater than 1 is either a prime number itself or can be represented as the product of prime numbers, and that this representation is unique, up to the order of the factors.

Understanding prime factorization is crucial for various number theory problems, such as finding the greatest common divisor (GCD), the least common multiple (LCM), and determining the number of divisors a number has (which is different from the number of *unique* prime divisors).

For instance, to find the total number of divisors (including 1 and the number itself) for 960 (\(2^6 \times 3^1 \times 5^1\)), you add 1 to each exponent and multiply the results: \((6+1) \times (1+1) \times (1+1) = 7 \times 2 \times 2 = 28\). So, 960 has 28 divisors in total, but only 3 unique prime divisors.

The question specifically asked for the number of *unique* prime divisors, which are the distinct prime bases in the prime factorization.