To solve the problem, we need to find the number of triplets \((a, b, c)\) of positive integers that satisfy the equation given by the determinant:
\[
|(a^3 + 1, a^2b, a^2c), (ab^2, b^3 + 1, b^2c), (ac^2, bc^2, c^3 + 1)| = 30
\]
### Step 1: Simplifying the Determinant
We can simplify the determinant by factoring out \(a\), \(b\), and \(c\) from the respective columns. This gives us:
\[
abc \cdot |(a^2 + \frac{1}{a}, ab, ac), (b^2, b^2 + \frac{1}{b}, bc), (c^2, bc, c^2 + \frac{1}{c})| = 30
\]
### Step 2: Rewriting the Determinant
Now we can rewrite the determinant as:
\[
|(a^2 + \frac{1}{a}, ab, ac), (b^2, b^2 + \frac{1}{b}, bc), (c^2, bc, c^2 + \frac{1}{c})|
\]
### Step 3: Elementary Row Operations
Next, we can perform elementary row operations to simplify the determinant further. We will add the rows together to create a simpler form:
1. Add the three rows together.
2. This results in a new first row of the form:
\[
(a^2 + b^2 + c^2 + 1, ab + ab + ac, ac + bc + c^2 + 1)
\]
### Step 4: Factor Out Common Terms
We can factor out \(a^2 + b^2 + c^2 + 1\) from the first row, which leads us to:
\[
(a^2 + b^2 + c^2 + 1) \cdot |(1, 1, 1), (b^2, b^2 + 1, b^2), (c^2, c^2, c^2 + 1)| = 30
\]
### Step 5: Evaluating the Simplified Determinant
Now we can evaluate the determinant on the right. The determinant simplifies to:
\[
(a^2 + b^2 + c^2 + 1) \cdot (1 \cdot (b^2(c^2 + 1) - b^2(c^2)) - 1 \cdot (b^2 - b^2)) = 30
\]
This gives us:
\[
(a^2 + b^2 + c^2 + 1) = 30
\]
### Step 6: Solving for Triplets
Now we need to solve the equation:
\[
a^2 + b^2 + c^2 = 29
\]
### Step 7: Finding Positive Integer Solutions
We need to find all combinations of positive integers \(a\), \(b\), and \(c\) such that their squares sum to 29.
1. **Possible values for \(a\)**:
- \(a = 1 \Rightarrow b^2 + c^2 = 28\)
- \(a = 2 \Rightarrow b^2 + c^2 = 25\)
- \(a = 3 \Rightarrow b^2 + c^2 = 20\)
- \(a = 4 \Rightarrow b^2 + c^2 = 13\)
- \(a = 5 \Rightarrow b^2 + c^2 = 4\)
- \(a = 6 \Rightarrow b^2 + c^2 = -7\) (not possible)
2. **Finding pairs \((b, c)\)** for each case:
- For \(a = 1\): Possible pairs are \((5, 1)\), \((1, 5)\), \((4, 4)\).
- For \(a = 2\): Possible pairs are \((5, 0)\), \((0, 5)\) (not valid as they must be positive).
- For \(a = 3\): Possible pairs are \((4, 2)\), \((2, 4)\), \((3, 3)\).
- For \(a = 4\): Possible pairs are \((3, 2)\), \((2, 3)\), \((1, 3)\), \((3, 1)\).
- For \(a = 5\): Possible pairs are \((2, 0)\), \((0, 2)\) (not valid).
### Step 8: Counting Unique Triplets
Count the unique permutations of the valid triplets:
- From \((1, 5, 1)\): 3 permutations.
- From \((3, 4, 2)\): 6 permutations.
- From \((4, 4, 1)\): 3 permutations.
### Conclusion
The total number of unique triplets \((a, b, c)\) that satisfy the equation is \(3 + 6 + 3 = 12\).
### Final Answer
The number of triplets \((a, b, c)\) of positive integers satisfying the equation is equal to **12**.
