Reduce each number to its digital root, do the same operation on the roots, and compare with the answer's root — a mismatch means an error.

The digital root is the digit sum reduced to one digit. To check 47 × 23 = 1081:

• root(47) = 4+7 = 11 → 2
• root(23) = 5
• 2 × 5 = 10 → 1
• root(1081) = 1+0+8+1 = 10 → 1 ✓ matches

Same for addition: 47 + 23 = 70; 2 + 5 = 7; root(70) = 7 ✓. If the two sides disagree, the answer is definitely wrong.

Because a number and its digit sum leave the same remainder when divided by 9 — the digital root is just the number mod 9.

Since 10 ≡ 1 (mod 9), every digit contributes only itself to the remainder, so any number ≡ its digit sum (mod 9). Arithmetic is preserved under "take the remainder mod 9": if a·b = c, then (a mod 9)·(b mod 9) ≡ c (mod 9). Casting out nines is exactly checking that congruence. (Digital roots run 1–9; a multiple of 9 gives 9, standing in for remainder 0.)