Find the number of ways in which 8064 can be resolved as the product of two factors?

Options

A. 20
B. 21
C. 22
D. 24

Answer

To find the number of ways in which 8064 can be resolved as the product of two factors, we will follow these steps:

### Step 1: Prime Factorization of 8064
First, we need to factor 8064 into its prime factors.

- Start by dividing 8064 by 2 (the smallest prime number):
- 8064 ÷ 2 = 4032
- 4032 ÷ 2 = 2016
- 2016 ÷ 2 = 1008
- 1008 ÷ 2 = 504
- 504 ÷ 2 = 252
- 252 ÷ 2 = 126
- 126 ÷ 2 = 63
- Now, 63 is not divisible by 2, so we divide by the next prime number, which is 3:
- 63 ÷ 3 = 21
- 21 ÷ 3 = 7
- Finally, 7 is a prime number.

Thus, the prime factorization of 8064 is:
\[ 8064 = 2^7 \times 3^2 \times 7^1 \]

### Step 2: Finding Total Number of Factors
To find the total number of factors of a number given its prime factorization \( p^a \times q^b \times r^c \), the formula is:
\[ (a+1)(b+1)(c+1) \]

For 8064:
- \( a = 7 \) (for \( 2^7 \))
- \( b = 2 \) (for \( 3^2 \))
- \( c = 1 \) (for \( 7^1 \))

Calculating the total number of factors:
\[ (7+1)(2+1)(1+1) = 8 \times 3 \times 2 \]

Now, calculate:
\[ 8 \times 3 = 24 \]
\[ 24 \times 2 = 48 \]

So, 8064 has a total of 48 factors.

### Step 3: Resolving into Products of Two Factors
To find the number of ways to express 8064 as a product of two factors, we need to consider that each pair of factors \( (a, b) \) such that \( a \times b = 8064 \) can be counted once.

Since each factorization \( (a, b) \) is the same as \( (b, a) \), we divide the total number of factors by 2:
\[ \text{Number of ways} = \frac{\text{Total Factors}}{2} = \frac{48}{2} = 24 \]

### Final Answer
Thus, the number of ways in which 8064 can be resolved as the product of two factors is **24**.

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