Find the number of divisors of the number 36000.

Answer

To find the number of divisors of the number 36,000, we will follow these steps:

### Step 1: Prime Factorization of 36,000
We start by performing the prime factorization of 36,000.

1. **Break down 36,000**:
\[ 36,000 = 36 \times 1,000 \]

2. **Factor 36**:
\[ 36 = 6 \times 6 = (2 \times 3) \times (2 \times 3) = 2^2 \times 3^2 \]

3. **Factor 1,000**:
\[ 1,000 = 10^3 = (2 \times 5)^3 = 2^3 \times 5^3 \]

4. **Combine the factors**:
\[ 36,000 = 36 \times 1,000 = (2^2 \times 3^2) \times (2^3 \times 5^3) \]

5. **Combine like bases**:
\[ 36,000 = 2^{2+3} \times 3^2 \times 5^3 = 2^5 \times 3^2 \times 5^3 \]

### Step 2: Use the Formula for Divisors
The formula to find the number of divisors \(d(n)\) of a number \(n = p_1^{e_1} \times p_2^{e_2} \times p_3^{e_3}\) is:
\[ d(n) = (e_1 + 1)(e_2 + 1)(e_3 + 1) \]
where \(e_1, e_2, e_3\) are the powers of the prime factors.

1. **Identify the powers**:
- For \(2^5\), the power is \(5\).
- For \(3^2\), the power is \(2\).
- For \(5^3\), the power is \(3\).

2. **Apply the formula**:
\[ d(36000) = (5 + 1)(2 + 1)(3 + 1) \]
\[ = 6 \times 3 \times 4 \]

### Step 3: Calculate the Result
1. **Calculate the product**:
\[ 6 \times 3 = 18 \]
\[ 18 \times 4 = 72 \]

### Final Answer
The number of divisors of the number 36,000 is **72**.

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