Let d(n) denote the number of positive divisors of a positive integer n. Which of the following are correct? 1. d(5) = d(11) Select the correct answer using the code given below:
2. d(5).d(11) = d(55)
3. d(5) + d(11) = d(16)
1 and 2 only
Understanding the Number of Positive Divisors, d(n)
The function d(n) represents the number of positive divisors of a positive integer n. A divisor is a number that divides n evenly without leaving a remainder. To find d(n) for a small number, we can list all its positive divisors and count them.
Calculating d(n) for the Given Numbers
Let's calculate the number of positive divisors for each integer mentioned in the statements: 5, 11, 55, and 16.
- d(5): The positive divisors of 5 are 1 and 5. There are 2 divisors. So, \(d(5) = 2\).
- d(11): The positive divisors of 11 are 1 and 11. There are 2 divisors. So, \(d(11) = 2\).
- d(55): The positive divisors of 55 are 1, 5, 11, and 55. There are 4 divisors. So, \(d(55) = 4\).
- d(16): The positive divisors of 16 are 1, 2, 4, 8, and 16. There are 5 divisors. So, \(d(16) = 5\).
Now that we have calculated the number of positive divisors for each integer, let's evaluate the correctness of the given statements.
Analyzing the Correctness of Each Statement on d(n)
We will check each statement using the values we calculated:
- Statement 1: \(d(5) = d(11)\)
- We calculated \(d(5) = 2\) and \(d(11) = 2\).
- Is \(2 = 2\)? Yes, this is true.
- Statement 1 is correct.
- Statement 2: \(d(5) \cdot d(11) = d(55)\)
- We calculated \(d(5) = 2\), \(d(11) = 2\), and \(d(55) = 4\).
- Let's check if the equation holds: \(2 \cdot 2 = 4\).
- Is \(4 = 4\)? Yes, this is true.
- Statement 2 is correct. This statement illustrates a property of the divisor function for coprime numbers. Since 5 and 11 are prime numbers and thus coprime (their greatest common divisor is 1), the number of divisors of their product is the product of their number of divisors.
- Statement 3: \(d(5) + d(11) = d(16)\)
- We calculated \(d(5) = 2\), \(d(11) = 2\), and \(d(16) = 5\).
- Let's check if the equation holds: \(2 + 2 = 5\).
- Is \(4 = 5\)? No, this is false.
- Statement 3 is incorrect.
Conclusion on Correct Statements
Based on our analysis, Statement 1 (\(d(5) = d(11)\)) and Statement 2 (\(d(5) \cdot d(11) = d(55)\)) are correct, while Statement 3 (\(d(5) + d(11) = d(16)\)) is incorrect.
Therefore, the correct answer is the one indicating that only statements 1 and 2 are correct.
Revision Table: Summary of Divisor Calculations and Statement Checks
| Integer (n) | Positive Divisors of n | Number of Divisors (d(n)) | Statement Checked | Evaluation | Correct? |
|---|---|---|---|---|---|
| 5 | 1, 5 | d(5) = 2 | Statement 1: \(d(5) = d(11)\) | \(2 = 2\) | Correct |
| 11 | 1, 11 | d(11) = 2 | |||
| 55 | 1, 5, 11, 55 | d(55) = 4 | Statement 2: \(d(5) \cdot d(11) = d(55)\) | \(2 \cdot 2 = 4\) | Correct |
| 16 | 1, 2, 4, 8, 16 | d(16) = 5 | Statement 3: \(d(5) + d(11) = d(16)\) | \(2 + 2 = 5\) | Incorrect |
Additional Information: The Divisor Function d(n)
The divisor function d(n), also often denoted as \(\tau(n)\) or \(\sigma_0(n)\), is an important function in number theory. It counts the number of positive divisors of a positive integer n.
A more general way to calculate d(n) for any positive integer n is using its prime factorization. If the prime factorization of n is given by \(n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}\), where \(p_1, p_2, \ldots, p_k\) are distinct prime numbers and \(a_1, a_2, \ldots, a_k\) are positive integers, then the number of positive divisors of n is given by the formula:
\[d(n) = (a_1 + 1)(a_2 + 1)\cdots(a_k + 1)\]
Let's apply this formula to the numbers in the question:
- For n=5: \(5 = 5^1\). Here \(p_1=5\), \(a_1=1\). So \(d(5) = (1+1) = 2\).
- For n=11: \(11 = 11^1\). Here \(p_1=11\), \(a_1=1\). So \(d(11) = (1+1) = 2\).
- For n=55: \(55 = 5^1 \cdot 11^1\). Here \(p_1=5\), \(a_1=1\), \(p_2=11\), \(a_2=1\). So \(d(55) = (1+1)(1+1) = 2 \cdot 2 = 4\).
- For n=16: \(16 = 2^4\). Here \(p_1=2\), \(a_1=4\). So \(d(16) = (4+1) = 5\).
These results match the results we got by listing divisors, confirming the formula's correctness.
The property observed in Statement 2, \(d(mn) = d(m)d(n)\) when \(\gcd(m, n) = 1\), means that the divisor function d(n) is a multiplicative function. This is a significant property in number theory and holds when m and n are coprime (have no common positive divisors other than 1).
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