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DS Digital Root Check

Reduce both sides of any calculation to a single digit. If the two digits disagree, the answer is definitely wrong — and you find out in two seconds, without redoing the work.

2 min readLevel ★Used in every other lesson

1. The rule

  1. Add up the digits of a number. If the sum has more than one digit, add again.
  2. Stop at one digit — that digit is the digital root (DS). Example: 4312 → 4+3+1+2 = 10 → 1+0 = 1.
  3. Do it for both sides of the operation. For × and +, the DS of the result must equal the DS of the DS-combination; for − and ÷ it works the same way.
  4. Compare. Different → the answer is wrong, always. Same → the answer passes this one check (it can still be wrong in a way that preserves the DS).

One-line version

Reduce both sides to one digit and compare. A mismatch is a proof of error; a match is only a green light, not a proof of correctness.

Because only one digit survives, the check costs two seconds and catches roughly eight out of nine careless mistakes: dropped carries, shifted place values, wrong multiplication table entries.

2. Worked example

Is 4312 × 42 = 181104 correct?

  • DS(4312) = 4+3+1+2 = 10 → 1
  • DS(42) = 4+2 = 6
  • Combine: 1 × 6 = 6
  • DS(181104) = 1+8+1+1+0+4 = 15 → 6 ✓ consistent

Now the same product written wrongly as 181108: DS = 1+8+1+1+0+8 = 19 → 1 ≠ 6, so the answer is rejected in one step.

3. Live demo

Type an expression and see both digital roots at once

The demo is computed in your browser; nothing is sent anywhere.

4. Practice

Apply the DS check only — do not compute the products in full.

  1. 4312 × 42 → 181104
  2. 47 × 63 → 2961
  3. 316 × 12 → 3792 and 316 × 12 → 3782: which one is impossible?
  4. Digital root of 12345678
  5. 5525 ÷ 25 → 221 or 223?
Answers & reasoning
1) DS 1 × 6 = 6 vs DS 15 → 6 ✓ consistent.   2) 2 × 9 = 18 → 9 vs DS(2961) = 18 → 9 ✓ consistent.   3) 1 × 3 = 3 vs DS(3782) = 20 → 2 → 3782 is impossible.   4) 1+2+3+4+5+6+7+8 = 36 → 9.   5) DS(5525) = 17 → 8 and DS(25) = 7; the quotient must satisfy 7 × q ≡ 8, so q → 5. DS(221) = 5 ✓, DS(223) = 7 ✗ → the answer is 221 (25 × 221 = 5525).
Practise this mode Next: Multiplying by 11 →

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