Casting Out Nines — Definition, Formula & Examples

Casting out nines is a method for checking whether an arithmetic answer is correct by replacing each number with its digital root (the repeated sum of its digits) and verifying that the same operation on those roots gives a consistent result.

Casting out nines exploits the fact that any integer is congruent to the sum of its digits modulo 9. If $a \circ b = c$ for an operation $\circ$ (addition, subtraction, or multiplication), then the digit sums must satisfy the same relationship modulo 9. A mismatch guarantees an error; a match suggests (but does not prove) correctness.

How It Works

To cast out nines, add up all the digits of a number. If the result has more than one digit, add those digits again until you get a single digit from 1 to 9 (treating 9 as 0). Do this for every number in your calculation. Then perform the same operation on those single-digit results and reduce again. If the final single digit does not match the digit sum of your original answer, you made an error somewhere.

Worked Example

Problem: Check whether 248 × 37 = 9176 using casting out nines.

Step 1: Find the digit sum of 248: 2 + 4 + 8 = 14, then 1 + 4 = 5.

248 \to 2+4+8 = 14 \to 1+4 = 5

Step 2: Find the digit sum of 37: 3 + 7 = 10, then 1 + 0 = 1.

37 \to 3+7 = 10 \to 1+0 = 1

Step 3: Multiply the two digit sums: 5 × 1 = 5.

5 \times 1 = 5

Step 4: Find the digit sum of the claimed answer 9176: 9 + 1 + 7 + 6 = 23, then 2 + 3 = 5.

9176 \to 9+1+7+6 = 23 \to 2+3 = 5

Step 5: Compare: both results are 5, so the answer is consistent. (The true product is indeed 9176.)

5 = 5 \checkmark

Answer: The digit sums match (both equal 5), so 248 × 37 = 9176 passes the check.

Why It Matters

Before calculators were common, casting out nines was the standard way students and accountants verified long multiplication and addition by hand. It remains a useful introduction to modular arithmetic, a topic central to number theory and computer science.

Common Mistakes

Mistake: Assuming a matching digit sum proves the answer is correct.

Correction: Casting out nines can only catch errors — it cannot confirm correctness. For example, swapping two digits (like writing 9167 instead of 9176) produces the same digit sum, so that mistake would slip through.