In how many ways can 1146600 be written as the product of two factors?

Options

A. 100
B. 108
C. 216
D. 273

Answer

To determine how many ways the number 1146600 can be expressed as the product of two factors, we will follow these steps:

### Step 1: Prime Factorization
First, we need to find the prime factorization of 1146600. We can do this by dividing the number by prime numbers until we reach 1.

1. Divide by 2:
- 1146600 ÷ 2 = 573300
- 573300 ÷ 2 = 286650
- 286650 ÷ 2 = 143325 (stop here as 143325 is odd)

2. Divide by 3:
- 143325 ÷ 3 = 47775
- 47775 ÷ 3 = 15925 (stop here as 15925 is not divisible by 3)

3. Divide by 5:
- 15925 ÷ 5 = 3185
- 3185 ÷ 5 = 637

4. Divide by 7:
- 637 ÷ 7 = 91

5. Divide by 13:
- 91 ÷ 13 = 7
- 7 ÷ 7 = 1

Thus, the prime factorization of 1146600 is:
\[ 1146600 = 2^3 \times 3^2 \times 5^2 \times 7^1 \times 13^1 \]

### Step 2: Calculate the Total Number of Factors
To find the total number of factors, we use the formula:
\[ \text{Total number of factors} = (e_1 + 1)(e_2 + 1)(e_3 + 1)(e_4 + 1)(e_5 + 1) \]
where \(e_i\) are the powers of the prime factors.

From our prime factorization:
- For \(2^3\), \(e_1 = 3\) → \(3 + 1 = 4\)
- For \(3^2\), \(e_2 = 2\) → \(2 + 1 = 3\)
- For \(5^2\), \(e_3 = 2\) → \(2 + 1 = 3\)
- For \(7^1\), \(e_4 = 1\) → \(1 + 1 = 2\)
- For \(13^1\), \(e_5 = 1\) → \(1 + 1 = 2\)

Now, we multiply these results:
\[ \text{Total number of factors} = 4 \times 3 \times 3 \times 2 \times 2 \]

Calculating this step-by-step:
1. \(4 \times 3 = 12\)
2. \(12 \times 3 = 36\)
3. \(36 \times 2 = 72\)
4. \(72 \times 2 = 144\)

So, the total number of factors of 1146600 is 144.

### Step 3: Calculate the Number of Ways to Write as Product of Two Factors
To find how many ways we can express 1146600 as the product of two factors, we divide the total number of factors by 2:
\[ \text{Ways to write as product of two factors} = \frac{\text{Total number of factors}}{2} = \frac{144}{2} = 72 \]

### Final Answer
Thus, the number of ways to write 1146600 as the product of two factors is **72**.

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