Understanding the Problem
To find the number of positive integral solutions for the equation \( xy = 24 \), we need to determine the factor pairs of 24, since each pair will represent a solution \((x, y)\).
Finding Factors of 24
First, we can find the prime factorization of 24:
- \( 24 = 2^3 \times 3^1 \)
Next, we can determine the number of factors:
- The formula to calculate the total number of factors from prime factorization \( p_1^{e_1} \times p_2^{e_2} \) is \((e_1 + 1)(e_2 + 1)\).
Applying this to our factorization:
- For \( 2^3 \): \( e_1 = 3 \) → \( e_1 + 1 = 4 \)
- For \( 3^1 \): \( e_2 = 1 \) → \( e_2 + 1 = 2 \)
Now, multiply these results:
- Total number of factors = \( 4 \times 2 = 8 \)
Identifying Positive Integral Solutions
Each factor of 24 can be paired with another factor to satisfy \( xy = 24 \):
- The factor pairs are:
- \( (1, 24) \)
- \( (2, 12) \)
- \( (3, 8) \)
- \( (4, 6) \)
- \( (6, 4) \)
- \( (8, 3) \)
- \( (12, 2) \)
- \( (24, 1) \)
Conclusion
Thus, the total number of positive integral solutions \((x, y)\) for the equation \( xy = 24 \) is:
- 8 unique pairs of factors.
Therefore, the answer is 8.