In how many ways can 576 be expressed as the product of two distinct factors?

Answer

To find the number of ways to express 576 as the product of two distinct factors, we can follow these steps:

### Step 1: Find the Prime Factorization of 576
First, we need to determine the prime factors of 576. We can do this by dividing 576 by the smallest prime numbers until we reach 1.

- 576 ÷ 2 = 288
- 288 ÷ 2 = 144
- 144 ÷ 2 = 72
- 72 ÷ 2 = 36
- 36 ÷ 2 = 18
- 18 ÷ 2 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1

Thus, the prime factorization of 576 is:
\[ 576 = 2^6 \times 3^2 \]

### Step 2: Calculate the Total Number of Factors
To find the total number of factors of a number from its prime factorization, we use the formula:
\[ (e_1 + 1)(e_2 + 1)(e_3 + 1) \]
where \( e_1, e_2, e_3 \) are the powers of the prime factors.

For 576:
- The power of 2 is 6, so \( 6 + 1 = 7 \)
- The power of 3 is 2, so \( 2 + 1 = 3 \)

Thus, the total number of factors is:
\[ 7 \times 3 = 21 \]

### Step 3: Count the Distinct Factor Pairs
Since we want to express 576 as a product of two distinct factors, we need to consider how many pairs of factors can be formed. Each pair consists of two factors \( (a, b) \) such that \( a \times b = 576 \) and \( a \neq b \).

To find the number of ways to form distinct pairs:
1. We have 21 total factors.
2. One of these factors is the square root of 576, which is \( 24 \) (since \( 24 \times 24 = 576 \)).
3. To find the distinct pairs, we subtract 1 (to exclude the square root) and divide by 2 (since each pair \( (a, b) \) is counted twice as \( (b, a) \)).

So, the calculation is:
\[ \frac{21 - 1}{2} = \frac{20}{2} = 10 \]

### Conclusion
Therefore, 576 can be expressed as the product of two distinct factors in **10 different ways**.

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