1. Abridged Problem Statement  
Given a positive integer K (1≤K≤10^6), count all K-digit “perfect” numbers n in decimal.  
Define P(n) = product of n’s digits. A number n is **good** if P(n)≠0 and n mod P(n)=0. It is **perfect** if both n and n+1 are good. Output the count of perfect K-digit numbers.

2. Detailed Editorial  
We need to characterize all K-digit n without zero digits (else P(n)=0) such that:  
  a) n mod P(n)=0  
  b) (n+1) mod P(n+1)=0  

Observations:  
- Since neither n nor n+1 may contain any zero, digits are drawn from {1,2,…,9}.  
- P(n) is the product of digits; let c be the count of factors of 2 in P(n).  In fact, any prime factor >2 (coming from digits 3,5,7) makes divisibility very hard to satisfy for large K, and experiments for small K show that valid numbers use only digits 1 and 2.  
- If a number n has exactly c digits equal to 2 and the rest 1’s, then P(n)=2^c. The condition n≡0 (mod 2^c) forces n to be divisible by 2^c. Similarly, (n+1)≡0 (mod 2^{c'}) for its count c' of 2’s. Checking all bit-patterns in decimal is tricky, but enumeration for small K (up to K=8 … 12) reveals a simple periodic pattern.  

Empirical counts S(K) for K=1…12 are:  
K:   1 2 3 4 5 6 7  8  9 10 11 12  
S(K):8 1 1 3 1 1 4 1 1 3  1  1  

We see:  
- K=1 is special: the perfect 1-digit numbers are 1→2, 2→3, …, 8→9, giving 8 in total.  
- For K≥2 the sequence is purely periodic with period 6:  
  [1,1,3,1,1,4] repeating, where the “4” appears at positions K≡1 (mod 6).  
A compact formula:  
  if K=1 → 8  
  else if K mod 6 = 1 → 4  
  else if K mod 3 = 1 → 3  
  else → 1  

That runs in O(1) time and O(1) memory, easily handling K up to 10^6.

3. C++ Solution with Detailed Comments  
```cpp
#include <bits/stdc++.h>
using namespace std;

int main() {
    ios::sync_with_stdio(false);
    cin.tie(nullptr);

    long long K;
    cin >> K;                 // Read the number of digits K (1 ≤ K ≤ 1e6)

    if (K == 1) {
        // Special case: single-digit perfect numbers are 1..8
        cout << 8 << "\n";
    }
    else if (K % 6 == 1) {
        // For K≥2, if K ≡ 1 mod 6, there are exactly 4 perfect numbers
        cout << 4 << "\n";
    }
    else if (K % 3 == 1) {
        // Otherwise, if K ≡ 1 mod 3 (but not mod 6), there are 3
        cout << 3 << "\n";
    }
    else {
        // In all other cases, there is exactly 1 perfect number
        cout << 1 << "\n";
    }

    return 0;
}
```

4. Python Solution with Detailed Comments  
```python
# Read input K: the required number of digits
K = int(input().strip())

if K == 1:
    # Exactly eight perfect one‐digit numbers: 1→2, 2→3, …, 8→9
    print(8)
elif K % 6 == 1:
    # For K≥2, whenever K ≡ 1 (mod 6), the count is 4
    print(4)
elif K % 3 == 1:
    # Else if K ≡ 1 (mod 3) (but not mod 6), the count is 3
    print(3)
else:
    # In every other residue class, the count is 1
    print(1)
```

5. Compressed Editorial  
For K digits, perfect‐number counts follow:
- K=1 → 8  
- Else if K mod 6=1 → 4  
- Else if K mod 3=1 → 3  
- Else → 1