Casting Out Sevens

A method for verifying the correctness of an arithmetical operation on natural numbers, based on the same principle as casting out nines. The methods of sevens takes advantage of the fact that the residue (mod 7) of a sum (or product) must be equal to the sum (or product) of the residues of the summands (or factors).

For example, the correct sum

(1)

corresponds to a correct sum of residues mod 7

(2)

where, on the right-hand side, 9 has been replaced by its residue 2 (mod 7).

On the other hand, the incorrect sum

(3)

gives rise to an incorrect sum of residues

(4)

since the right-hand side should be 0.

Tests based on the comparison of residues are not completely reliable since they leave some errors undetected (namely, an incorrect sum can produce a correct sum of residues). Hence it can be helpful to double-check with respect to 7 and 9.