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Parity Addition of several numbers

Suppose we want to parity add several numbers. We can proceed in a natural way: add first two, then add the third one to the result, then add the forth one to the last result and so on. As there is no carry on, we perform all additions by columns. That means that we can parity add several numbers by columns too. Hence, all the properties of parity addition of several numbers are seen when we use one digit numbers. Let us make a table for parity adding three different one-digit numbers. We consider all possible cases in the table below:

ABCA oplus B(A oplus B) oplus C

Looking at this table see that the result is 1 if we parity add one or three ones; and the result is zero if we parity add zero or two ones. The following simple rule is true: the result is 1 if we add odd number of ones, and 0 if we add even number of ones.

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