The standard formula for the sum of all numbers formed by permuting m distinct digits is:
Sum = ( Sum of the digits ) × (m - 1)! × ( 111… 1 repeated m times )
For a 4-digit setup (m = 4):
Sum = (a + b + c + d) × 3! × 1111 = 6666(a + b + c + d)
We are given that this total value equals 153310 + n:
6666(a + b + c + d) = 153310 + n
Since n is a single-digit natural number (1 ≤ n ≤ 9), the total sum must be a perfect multiple of 6666 that is slightly larger than 153310. Divide 153310 by 6666 to find the multiplier:
(153310)/(6666) ≈ 22.9988
The nearest higher integer multiple is 23. Let us evaluate this target:
6666 × 23 = 153318
This satisfies the equation:
153310 + n = 153318 ⇒ n = 8
The base digit sum is:
a + b + c + d = 23
Final calculated value = (a + b + c + d) + n = 23 + 8 = 31
