The number of divisors of 441, 1125 and 384 are in

Options

A. A.P
B. G.P
C. H.P
D. none of these

Answer

To find the number of divisors of the numbers 441, 1125, and 384, we will follow these steps:

### Step 1: Prime Factorization of Each Number

1. **Factor 441**:
- 441 can be factored as \(21 \times 21\) or \(3^2 \times 7^2\).
- Thus, the prime factorization is \(3^2 \times 7^2\).

2. **Factor 1125**:
- 1125 can be factored as \(5^3 \times 3^2\).
- Thus, the prime factorization is \(5^3 \times 3^2\).

3. **Factor 384**:
- 384 can be factored as \(2^7 \times 3^1\).
- Thus, the prime factorization is \(2^7 \times 3^1\).

### Step 2: Apply the Divisor Formula

The formula to find the number of divisors \(d(n)\) of a number \(n = p_1^{a_1} \times p_2^{a_2} \times ... \times p_k^{a_k}\) is given by:

\[ d(n) = (a_1 + 1)(a_2 + 1)...(a_k + 1) \]

1. **Calculate the number of divisors for 441**:
- From \(3^2 \times 7^2\), we have \(a_1 = 2\) and \(a_2 = 2\).
- Therefore, \(d(441) = (2 + 1)(2 + 1) = 3 \times 3 = 9\).

2. **Calculate the number of divisors for 1125**:
- From \(5^3 \times 3^2\), we have \(a_1 = 3\) and \(a_2 = 2\).
- Therefore, \(d(1125) = (3 + 1)(2 + 1) = 4 \times 3 = 12\).

3. **Calculate the number of divisors for 384**:
- From \(2^7 \times 3^1\), we have \(a_1 = 7\) and \(a_2 = 1\).
- Therefore, \(d(384) = (7 + 1)(1 + 1) = 8 \times 2 = 16\).

### Step 3: Summary of Results

- The number of divisors of:
- 441 is **9**.
- 1125 is **12**.
- 384 is **16**.

### Step 4: Conclusion

Now we can conclude that the number of divisors of 441, 1125, and 384 are 9, 12, and 16 respectively.

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