The digits of the three-digit integers $a, b,$ and $c$ are the nine nonzero digits $1,2,3,\cdots 9$ each of them appearing exactly once. Given that the ratio $a:b:c$ is $1:3:5$, determine $a, b,$ and $c$.
1. **Identify the constraints and initial conditions:**
   - The digits of the three-digit integers \(a\), \(b\), and \(c\) are the nine nonzero digits \(1, 2, 3, \ldots, 9\), each appearing exactly once.
   - The ratio \(a:b:c\) is \(1:3:5\).

2. **Determine the possible values for \(a\):**
   - Since \(a\) is the smallest number in the ratio \(1:3:5\), its first digit must be small enough to ensure \(5a\) is still a three-digit number.
   - If the first digit of \(a\) were \(2\) or higher, \(5a\) would exceed three digits. Therefore, the first digit of \(a\) must be \(1\).

3. **Determine the possible values for \(c\):**
   - Since \(c = 5a\), the last digit of \(c\) must be \(5\) because there is no \(0\) in the list of digits.

4. **Casework on \(b\):**
   - The first digit of \(b\) could be \(3\) or \(4\) (it cannot be \(5\) because \(5\) is already used in \(c\)).
   - We start with the first digit of \(b\) being \(3\).

5. **List possible values for \(b\) and corresponding \(a\):**
   - The digits available are \(2, 4, 6, 7, 8, 9\).
   - Possible values for \(b\) that are divisible by \(3\) are: \(324, 327, 342, 348, 369, 372, 378, 384, 387\).
   - Corresponding values for \(a\) (since \(b = 3a\)) are: \(108, 109, 114, 116, 123, 124, 126, 128, 129\).

6. **Eliminate invalid values for \(a\):**
   - \(a\) cannot be even because \(5a\) would end in \(0\), which is not allowed.
   - Eliminate even numbers and numbers containing \(0\): \(108, 109, 114, 116, 124, 126, 128\).
   - Remaining candidates for \(a\) are \(123\) and \(129\).

7. **Check for valid combinations:**
   - For \(a = 123\), \(b = 369\), but digits are repeated.
   - For \(a = 129\), \(b = 387\), and \(c = 645\), all digits are unique and non-repeating.

8. **Verify the solution:**
   - \(a = 129\), \(b = 387\), \(c = 645\).
   - Check the ratio: \(\frac{b}{a} = \frac{387}{129} = 3\) and \(\frac{c}{a} = \frac{645}{129} = 5\).
   - All conditions are satisfied.

The final answer is \( \boxed{ (a, b, c) = (129, 387, 645) } \).