How many factors are there of the number 2520?

Answer

To find the number of factors of the number 2520, we will follow these steps:

### Step 1: Prime Factorization of 2520
First, we need to factorize the number 2520 into its prime factors.

1. **Divide by 2**:
- 2520 ÷ 2 = 1260
- 1260 ÷ 2 = 630
- 630 ÷ 2 = 315 (315 is not divisible by 2 anymore)

2. **Divide by 3**:
- 315 ÷ 3 = 105
- 105 ÷ 3 = 35 (35 is not divisible by 3 anymore)

3. **Divide by 5**:
- 35 ÷ 5 = 7 (7 is a prime number)

So, the prime factorization of 2520 is:
\[ 2520 = 2^3 \times 3^2 \times 5^1 \times 7^1 \]

### Step 2: Use the Formula for Finding the Number of Factors
The formula to find the total number of factors from the prime factorization is:
If \( n = p_1^{e_1} \times p_2^{e_2} \times p_3^{e_3} \times \ldots \times p_k^{e_k} \), then the number of factors \( T(n) \) is given by:
\[ T(n) = (e_1 + 1)(e_2 + 1)(e_3 + 1) \ldots (e_k + 1) \]

### Step 3: Apply the Formula
From our prime factorization:
- For \( 2^3 \): \( e_1 = 3 \) → \( e_1 + 1 = 4 \)
- For \( 3^2 \): \( e_2 = 2 \) → \( e_2 + 1 = 3 \)
- For \( 5^1 \): \( e_3 = 1 \) → \( e_3 + 1 = 2 \)
- For \( 7^1 \): \( e_4 = 1 \) → \( e_4 + 1 = 2 \)

Now, we can calculate the total number of factors:
\[ T(2520) = (3 + 1)(2 + 1)(1 + 1)(1 + 1) \]
\[ T(2520) = 4 \times 3 \times 2 \times 2 \]

### Step 4: Calculate the Result
Now we perform the multiplication:
\[ T(2520) = 4 \times 3 = 12 \]
\[ T(2520) = 12 \times 2 = 24 \]
\[ T(2520) = 24 \times 2 = 48 \]

Thus, the total number of factors of 2520 is **48**.

### Summary
The number of factors of 2520 is **48**.

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