Add the digits, and keep adding until one digit is left. For 9875: 9+8+7+5 = 29, then 2+9 = 11, then 1+1 = 2. The shortcut 1 + (n − 1) mod 9 gives the same 2 in one step, because ten leaves a remainder of one when divided by nine.
Why it is always mod 9
The reason is one fact: 10 leaves a remainder of 1 when divided by 9. So does 100, and 1,000, and every power of ten.
That means 300 and 3 leave the same remainder mod 9, and 40 and 4 do, and so on. Adding the digits is therefore not an approximation of the number — it is a number with the same remainder mod 9, which is why repeating it converges to something meaningful rather than to noise.
The only wrinkle is that the remainders run 0 to 8 while digital roots run 1 to 9. A multiple of nine has remainder 0 and digital root 9, not 0. That is what the “1 + (n − 1)” in the formula is fixing.
Zero is the exception: its digital root is 0, and it is the only number for which that is true.
Divisibility falls straight out
Because the digital root shares a remainder mod 9 with the number, two divisibility tests come for free.
Divisible by 9 exactly when the digital root is 9.
Divisible by 3 exactly when the digital root is 3, 6 or 9 — because 3 divides 9, so the same congruence works one level down.
There is no equivalent for 7 or 11 by this route. 10 is not congruent to 1 modulo either of them, so the digits do not sum in a useful way — the 11 test alternates signs instead, which is a different trick from the same family.
Casting out nines
An old and genuinely useful way to check arithmetic done by hand.
Take the digital root of each input, do the same operation on those small numbers, take the digital root of the result, and compare it with the digital root of the answer you got. If they disagree, the answer is wrong.
It works because digital roots respect addition and multiplication — the root of a sum is the root of the sum of the roots. So the whole calculation can be shadowed by a much smaller one running alongside it.
Before calculators this was standard practice in bookkeeping, and a ledger page would carry the check in the margin.
What the check cannot catch
The test is one-directional, and being clear about which direction matters.
A mismatch is conclusive. If the roots disagree, the arithmetic is definitely wrong. No exceptions.
A match proves nothing. Roughly one wrong answer in nine has the right digital root by coincidence and sails through.
And one error type escapes every time: a transposition. Writing 21 where you meant 12 leaves the digit sum untouched, so the root is unchanged and the check passes. Transpositions are among the commonest copying errors there are, which is a real limitation rather than a footnote — and it is exactly why account numbers use a proper check digit instead.
The calculator above lets you feed it a wrong sum on purpose, so the failure mode is something you can see rather than something you have to take on trust.
Where it gets used
Checking hand arithmetic. Still the fastest sanity check on a long addition or multiplication, and it needs no equipment.
Divisibility. Deciding whether a large number is a multiple of three or nine without dividing it.
Recreational number theory. Digital roots turn up in magic squares, in patterns in the nine times table, and in the Vedic-mathematics tradition where they are called the “beejank”.
Check digits, by contrast. Modern schemes — ISBN, IBAN, the Luhn algorithm on a card number — weight the digits by position precisely so that transpositions do change the result. They are the answer to this page's limitation.
Sources and methodology
The result and its limits are elementary number theory; these are the references.
Method. Digital roots are computed on arbitrary-precision integers, so a hundred-digit number is handled exactly rather than through a double that would have lost its low digits long before. The repeated addition and the closed form are both computed and the suite checks them against each other on every number from 1 to 2,000 — a page that showed the working from one route and the answer from another could otherwise drift without anyone noticing. That engine is verified on every change against 140 hand-written assertions, including that repeated addition and the mod-9 shortcut agree on every number from 1 to 2,000, and that an off-by-one error is caught by casting out nines in all 3,600 generated cases while a transposition is not. The count and the per-case breakdown are published on the formula verification page.
Read the guide
The congruence the shortcut rests on — that ten is congruent to one modulo nine — is worked through on the Modulo Calculator.