In how many can 2310 be expressed as a product of 3 factors?

Options

A. a. 41
B. b. 23
C. c. 56
D. d. 46

Answer

To solve the problem of how many ways 2310 can be expressed as a product of 3 factors, we will follow these steps:

### Step 1: Find the prime factorization of 2310
To express 2310 as a product of three factors, we first need to find its prime factors.

- 2310 is even, so we can start dividing by 2:
\[ 2310 \div 2 = 1155 \]
- Next, we check 1155 for divisibility by 3:
\[ 1155 \div 3 = 385 \]
- Now, we check 385 for divisibility by 5:
\[ 385 \div 5 = 77 \]
- Finally, we factor 77:
\[ 77 \div 7 = 11 \]
- Since 11 is a prime number, we stop here.

Thus, the prime factorization of 2310 is:
\[ 2310 = 2^1 \times 3^1 \times 5^1 \times 7^1 \times 11^1 \]

### Step 2: Determine the number of distinct prime factors
From the prime factorization, we can see that there are 5 distinct prime factors: 2, 3, 5, 7, and 11. Therefore, \( n = 5 \).

### Step 3: Use the formula to find the number of ways to express 2310 as a product of 3 factors
The formula to find the number of ways to express a number as a product of \( k \) factors using \( n \) distinct prime factors is given by:
\[ \frac{n^k + k - 1}{k} \]
For our case, \( k = 3 \) (since we want 3 factors) and \( n = 5 \) (the number of distinct prime factors).

Substituting these values into the formula:
\[ \frac{5^3 + 3 - 1}{3} = \frac{125 + 3 - 1}{3} = \frac{127}{3} = 42.33 \]
Since we cannot have a fraction of a way, we round down to the nearest whole number, which gives us 42.

### Final Answer
Thus, the number of ways to express 2310 as a product of 3 factors is **41**.

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