How can you use a "digit sum check" modulo $\beta - 1$ or $\beta + 1$ to prove that an operation is faulty or incorrect? For example, given the expression below where base $\beta$ is 5:
$142\bar3_5 + 2\bar214_5 = 33\bar21_5$
How can I use a digit sum check with modulo to prove that the operation is incorrect? What are the steps in this process?
For $\beta=10$, this corresponds to forming remainders modulo $\beta-1=9$ and $\beta+1=11$, which are used in the well-known divisibility checks for those two divisors. The remainder modulo $\beta-1$ can be formed by summing the digits and iterating until a single digit remains. The remainder modulo $\beta+1$ can be formed by summing the digits with alternating signs (adding multiples of $\beta+1$ if the result is negative) and iterating until a value $\le\beta$ remains. In both cases, the result of the operation is found to be incorrect if the remainders on the two sides of the equation don't coincide.