Digit Sum Calculator – Digital Root & Checks

The digit sum simply adds a number’s digits: 987 gives 9+8+7 = 24. Iterating yields the digital root: 24 → 2+4 = 6. The digital root always equals n mod 9 (with 9 for multiples of 9), because 10 ≡ 1 (mod 9) — a fact that powers "casting out nines" arithmetic checks. Testing 123 + 459 = 582: digit sums 6, 18→9, and 15→6; since 6+9 ≡ 6 (mod 9), the addition passes. The calculator computes digit sum, digital root, and shows the mod-9 congruence for any input, plus automatic verification of sums and products you paste in. It also flags the classic trick extensions: a number is divisible by 3 exactly when its digit sum is (87 → 15 → divisible by 3), and by 9 when the root is 9.

1. 1

   Enter a number

   Any integer — 987 or 1,234,567

2. 2

   Sum the digits

   987 → 9+8+7 = 24, shown digit by digit

3. 3

   Iterate to the digital root

   24 → 6; the root equals n mod 9

4. 4

   Run a casting-out-nines check

   Verify pasted sums/products by comparing roots

• 1

  Digital root = n mod 9 (9 when n ≡ 0)

• 2

  Digit sums catch transpositions: 61 vs 16 differ by 45 → root change

• 3

  Mod-9 checks miss errors that are themselves multiples of 9

• Stopping at the digit sum (24) instead of iterating to the root (6)

• Using the check on subtraction without borrowing rules

• Assuming a passing cast-out-nines proves the answer correct

• Forgetting multiples of 9 have digital root 9, not 0