Question

If n = pqr, where p, q, and r are three different positive prime numbers, how many different positive divisors does n have, including 1 and n?

• A. 3
• B. 5
• C. 6
• D. 7
• E. 8

Hint:

The formula method is very powerful and much faster than listing, especially for numbers with higher powers in their prime factorization. For any number \(n = p^a q^b\), the number of divisors is \((a+1)(b+1)\). This is a fundamental concept in number theory and very useful for competitive exams.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This question is about number theory, specifically finding the total number of positive divisors (or factors) of a given number. The number is expressed as a product of three distinct prime numbers.
Step 2: Key Formula or Approach:
There are two common methods to solve this:
1. Listing Method: Systematically list all possible combinations of the prime factors.
2. Formula Method: If a number's prime factorization is \(n = p_1^{a_1} p_2^{a_2} \dots p_k^{a_k}\), then the total number of divisors is given by the product \((a_1 + 1)(a_2 + 1) \dots (a_k + 1)\).
Step 3: Detailed Explanation:
Method 1: Listing the Divisors
The number is \(n = pqr\). The divisors are formed by taking combinations of its prime factors \(p, q,\) and \(r\).
- Divisors with zero prime factors: 1. (1 divisor)
- Divisors with one prime factor: \(p, q, r\). (3 divisors)
- Divisors with two prime factors: \(pq, pr, qr\). (3 divisors)
- Divisors with three prime factors: \(pqr\) (which is \(n\)). (1 divisor)
The total number of divisors is the sum of these counts: \(1 + 3 + 3 + 1 = 8\).
Method 2: Using the Formula
The given number is \(n = pqr\). The prime factorization can be written with exponents as:
\[ n = p^1 q^1 r^1 \] The exponents of the prime factors are \(a_1 = 1\), \(a_2 = 1\), and \(a_3 = 1\).
Using the formula for the number of divisors:
\[ \text{Number of divisors} = (a_1 + 1)(a_2 + 1)(a_3 + 1) \] \[ \text{Number of divisors} = (1 + 1)(1 + 1)(1 + 1) \] \[ \text{Number of divisors} = 2 \times 2 \times 2 = 8 \] Both methods yield the same result. The number of divisors includes 1 and \(n\) itself.
Step 4: Final Answer:
The number \(n\) has 8 different positive divisors.