From inputs to Casting Out Nines output
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 identity used by Casting Out Nines
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
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.
Details to check in Casting Out Nines
Preserve First integer when checking the Casting Out Nines output. Keep Operation under the same convention and estimate Check result. A controlled change to Second integer should move the Casting Out Nines Check result in a mathematically consistent direction.
When to reach for Casting Out Nines
The method is a fast paper check for addition and multiplication. It catches many single-digit mistakes without repeating the original full calculation.
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.
Confirming the Casting Out Nines setup
Substitute the Casting Out Nines solution into First integer. This Casting Out Nines check rejects false Casting Out Nines branches and forbidden denominators.
Choose an easy First integer value before running Casting Out Nines. Predict Operation, then compare it with the Casting Out Nines output.
The scale of Check result can be challenged with a simpler First integer. Keep Operation unchanged during this Casting Out Nines trial. If the estimated Check result and calculated Check result differ sharply, revisit the entries before extending the Casting Out Nines work.
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.