What is the 6’s complement of (1543)7?

Concept: In base r, the (r − 1)'s complement (the diminished radix complement) is formed by replacing every digit d with (r − 1) − d. This is the direct generalization of the 1's complement you already know from binary, where r = 2 so r − 1 = 1; here r = 7, so r − 1 = 6 — hence the “6's complement.”

Application:

1. Digits of (1543)7 (left to right): 1, 5, 4, 3.

2. Since r − 1 = 6, replace each digit d with 6 − d.

3. 6 − 1 = 5

4. 6 − 5 = 1

5. 6 − 4 = 2

6. 6 − 3 = 3

7. Assembling the results in the same digit order gives (5123)7.

Cross-check: the (r − 1)'s complement is self-inverse — complementing the result should reproduce the original number. Applying 6 − d to each digit of 5, 1, 2, 3 gives 6 − 5 = 1, 6 − 1 = 5, 6 − 2 = 4, 6 − 3 = 3, i.e. (1543)7 — exactly the number we started with, confirming the computation.

Result: the 6's complement of (1543)7 is (5123)7.