1. The problem involves a number machine with two steps: first an operation \(An!\), then multiplication by 5, giving the output. 2. Given the input is 15 and the output is 30, we have:\n\[\text{Output} = (An!) \times 5 = 30\] 3. To find \(An!\) when input is 15, rearrange:\n\[An! = \frac{30}{5} = 6\] 4. Thus, the value after the first step \(An!\) for input 15 is 6. 5. But the notation \(An!\) suggests "factorial of some function of input". Since the input is 15 and the factorial is 6, it implies that \(An = 3\) because \(3! = 6\). 6. So \(An = 3\) when input is 15, or more generally, \(An = \frac{input}{5}\). 7. Check the machine formula: start with input \(x\), compute \(An = \frac{x}{5}\), then output is \(An! \times 5\). 8. So the complete number machine is: \nInput \(x\) \rightarrow \text{Calculate} \ An = \frac{x}{5}\ \rightarrow \text{Output} = An! \times 5\. Final answer: \[ \boxed{\text{Output} = \left(\frac{input}{5}\right)! \times 5} \] This completes the number machine description.