In how many ways 8100 can be resolved into product of two factors ?

Answer

To find the number of ways 8100 can be resolved into the product of two factors, we will follow these steps:

### Step 1: Prime Factorization of 8100
First, we need to find the prime factorization of 8100.

- 8100 can be broken down as follows:
- 8100 = 81 × 100
- 81 = 3^4 (since 81 = 3 × 3 × 3 × 3)
- 100 = 10^2 = (2 × 5)^2 = 2^2 × 5^2

Combining these, we get:
\[ 8100 = 2^2 × 3^4 × 5^2 \]

### Step 2: Identify the Exponents
From the prime factorization, we identify the exponents:
- \( k_1 = 2 \) (for the prime factor 2)
- \( k_2 = 4 \) (for the prime factor 3)
- \( k_3 = 2 \) (for the prime factor 5)

### Step 3: Check if 8100 is a Perfect Square
Next, we check if 8100 is a perfect square. A number is a perfect square if all the exponents in its prime factorization are even.

- \( k_1 = 2 \) (even)
- \( k_2 = 4 \) (even)
- \( k_3 = 2 \) (even)

Since all the exponents are even, 8100 is indeed a perfect square.

### Step 4: Use the Formula for Perfect Squares
For a perfect square, the number of ways to express it as a product of two factors is given by:
\[ \frac{(k_1 + 1)(k_2 + 1)(k_3 + 1)}{2} \]

Substituting the values of \( k_1, k_2, \) and \( k_3 \):
\[ \frac{(2 + 1)(4 + 1)(2 + 1)}{2} = \frac{3 \times 5 \times 3}{2} \]

### Step 5: Calculate the Result
Now we calculate:
\[ 3 \times 5 = 15 \]
\[ 15 \times 3 = 45 \]
\[ \frac{45}{2} = 22.5 \]

Since we can only have whole number factors, we round down to 22.

### Final Answer
Thus, the number of ways 8100 can be resolved into the product of two factors is:
\[ \boxed{23} \]