A numerical walkthrough
For 4,783 × 216, the residues are 4 and 0, so the expected residue is 0. The claimed product 1,033,128 also has residue 0 and passes the check.
Math calculator
Casting Out Nines Checker
Check a claimed addition or multiplication using digit residues modulo nine. A checkable formula accompanies check result instead of leaving an unexplained number.
Casting Out Nines Checker inputs
Known quantities
The idea behind the result
Casting out nines replaces each integer with its residue modulo 9 and checks whether the smaller operation has the same residue as the claimed result.
The calculation treats first integer, operation, second integer and claimed result in the roles printed beside each Casting Out Nines field for casting out nines checker.
Working through Casting Out Nines on paper
Reduce each operand by summing digits, perform the selected small operation, reduce again, and compare that residue with the claimed result. A hand-worked extension of Casting Out Nines Checker is exact product.
Details to check in Casting Out Nines
A pass is not proof: errors differing by a multiple of nine and digit transpositions may still pass. A failure does prove that the claimed result is wrong. If the Casting Out Nines Checker assumptions do not fit, consider digital roots.
Save extra digits internally if check result will become an input to another calculation.
When the result is useful
The method is a fast paper check for addition and multiplication. It catches many single-digit mistakes without repeating the original full calculation.
How the inputs shape the output
A field name is part of the formula. Match the problem's quantities to first integer, operation, second integer and claimed result, then check that they share the scale assumed by the casting out nines checker relationship.
Changing the claimed result by nine leaves the check unchanged, which explains both the method and its limitation. This is a stronger check than judging the answer only by how many decimal places it shows.
Match the reported precision to first integer, operation, second integer and claimed result, not to the number of digits the browser can display. Preserve an exact form when it communicates the casting out nines checker structure more clearly than a decimal.
This is an error screen, not a replacement for exact arithmetic. The formula panel makes the chosen definition explicit.
The minimum audit trail is short: first integer, operation, second integer and claimed result, their units, and the formula beside the answer. It is enough to distinguish this calculation from a similar-looking shortcut.
Questions about Casting Out Nines
Does passing guarantee the answer is correct?
No.
Why do transpositions pass?
Reordering digits does not change their sum modulo 9.
Can this check subtraction?
The underlying method can, but this page focuses on addition and multiplication.
What does a failure mean?
The claimed result is definitely inconsistent modulo 9.
Are negative integers allowed?
Yes; residues are normalized for comparison.