Every divisibility rule from 2 to 16, applied to your number and checked.
Which numbers divide it
Each rule checked by dividing.
3,168
Divisible by 2, 3, 4, 6, 8, 9, 11, 12, 16
Not divisible by 5, 7, 10, 13.
Digit sum
18
reduces to 9
Alternating sum
0
the one the rule for 11 uses
Rules that pass
9 of 13
each confirmed by dividing
Your divisor
—
enter one on the right
Every rule, applied and checked
Each divisibility rule with the working, the verdict, the true remainder and whether they agree
By
The rule
On this number
Remainder
2
The last digit is even.
Last digit 8 is even.
0
3
The digits add to a multiple of 3.
3 + 1 + 6 + 8 = 18 → 9, a multiple of 3.
0
4
The last two digits form a multiple of 4.
Last two digits are 68, a multiple of 4.
0
5
The last digit is 0 or 5.
Last digit is 8.
3
6
Divisible by both 2 and 3.
Even, and the digit sum is 18.
0
7
Double the last digit and subtract it from the rest; repeat.
316 − 2×8 = 300 → 30 − 2×0 = 30 → 30, not a multiple of 7
4
8
The last THREE digits form a multiple of 8.
Last three digits are 168, a multiple of 8.
0
9
The digits add to a multiple of 9.
3 + 1 + 6 + 8 = 18 → 9, a multiple of 9.
0
10
The last digit is 0.
Last digit is 8.
8
11
Alternate + and − across the digits from the right; the result is a multiple of 11.
−3 +1 −6 +8 = 0, a multiple of 11.
0
12
Divisible by both 3 and 4.
Digit sum 18, last two digits 68.
0
13
Multiply the last digit by 4 and ADD it to the rest; repeat.
316 + 4×8 = 348 → 34 + 4×8 = 66 → 66, not a multiple of 13
9
16
The last four digits form a multiple of 16.
Last four digits are 3168, a multiple of 16.
0
The remainder column comes from actually dividing. It is there so the rules can be checked rather than believed — several of them circulate in wrong versions.
Why each rule works
2: Ten is a multiple of 2, so every digit except the last contributes a multiple of 2. Only the units digit can break it.
3: Ten leaves remainder 1 when divided by 3, so every power of ten does too. Each digit therefore contributes just itself, and the whole number is congruent to the sum of its digits.
4: A hundred is a multiple of 4, so everything above the tens column is already divisible. Only the last two digits are left to decide it.
7: There is no clean digit rule for 7, because ten leaves remainder 3 rather than 0, 1 or −1. This subtraction is one of several equivalent tricks; for most people, dividing is genuinely faster.
8: A thousand is a multiple of 8, so everything above the hundreds column is already divisible. Three digits, not two — the two-digit version of this rule is simply wrong.
11: Ten leaves remainder −1 modulo 11, so the powers of ten alternate between +1 and −1. That is exactly what the alternating sum computes.
Nine of these rules are the same idea in different clothes: what matters is the remainder ten leaves on division by the divisor. It is 0 for 2, 5 and 10, which is why only the last digit counts; 1 for 3 and 9, which is why the digits can simply be added; and −1 for 11, which is why the signs alternate.
The rule for 7 exists mostly as a curiosity. Ten leaves remainder 3 modulo 7, which gives no tidy weighting, and the subtraction trick takes about as long as the division would.
Every rule above was also checked by actually dividing. They agree — as they must — but a page that only stated the rules would be asking you to take that on trust.
Every rule agrees with the division on this number.
What this tool shows
Nine of these thirteen rules are the same idea: what remainder ten leaves when divided by the divisor. Seeing that once explains the digit sum, the alternating sum and the last-digit rules together.
Rules for 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 and 16
The working for each, on your number
The true remainder, as a check on the rule
Why each rule holds
A custom divisor, checked by division
The rules that circulate in wrong versions
Rules checked by dividing The reason beside each rule Any custom divisor too The wrong versions corrected
Updated 7 September 2026 · Works in any browser, no installation
Almost every divisibility rule is the same observation: what remainder ten leaves when divided by the divisor. For 2, 5 and 10 it is zero, so only the last digit can matter. For 3 and 9 it is one, so every place has equal weight and the digits can simply be added. For 11 it is minus one, so the signs alternate. For 4, 8 and 16, a high enough power of ten is divisible, so a fixed number of trailing digits decides it.
At a glance
Formula shown
Each rule follows from the remainder 10 leaves modulo the divisor: 0 for 2, 5 and 10 (last digit only), 1 for 3 and 9 (digits may be summed), \u22121 for 11 (alternating sum), and 0 for higher powers of 10 modulo 2^k (last k digits).
Scenario support
Checking a number by hand before a long division; testing whether a total splits evenly; a quick sanity check on an arithmetic answer.
Educational estimate
Planning support from the values you enter — not professional advice.
Why the rules work at all
A number written in decimal is a sum of digits times powers of ten. So its remainder on division by d depends only on what remainders the POWERS OF TEN leave modulo d — and those follow a short pattern.
When ten leaves 0, every power above the units leaves 0 too, so only the last digit survives. That gives the rules for 2, 5 and 10 in one line.
When ten leaves 1, every power leaves 1, so each digit contributes exactly itself and the whole number is congruent to the sum of its digits. That gives 3 and 9.
When ten leaves −1, the powers alternate +1, −1, +1, and so does the weighting. That gives 11.
The digit sum, for 3 and 9
Ten leaves remainder 1 modulo both 3 and 9, so every power of ten does. Each digit therefore contributes just itself, and the number is divisible exactly when its digit sum is.
It can be repeated. 123456789 has digit sum 45, which has digit sum 9, so the original is divisible by both 3 and 9. The repeated sum is called the digital root, and for any number it equals the number modulo 9, with 9 in place of 0.
This is also the basis of casting out nines, the old check on hand arithmetic: if two numbers multiply, their digit sums multiply modulo 9 too. It catches most slips, though not a transposition — swapping two digits leaves the sum unchanged.
The alternating sum, for 11
Modulo 11, ten leaves −1. So a hundred leaves +1, a thousand −1, and the weights alternate all the way up.
Adding the digits with alternating signs, starting from the RIGHT, gives a number divisible by 11 exactly when the original is. For 1001: +1 −0 +0 −1 = 0, and 1001 = 7 × 11 × 13.
Direction matters less than consistency — starting from the left flips every sign, which changes the answer’s sign but not whether it is a multiple of 11. Starting from the right is the convention here.
Trailing digits, for 4, 8 and 16
A hundred is divisible by 4, so everything above the tens column already is — only the last two digits can decide it. A thousand is divisible by 8, so the last three decide that one. Ten thousand is divisible by 16, so four digits.
The rule for 8 is three digits, not two. The two-digit version circulates widely and is simply wrong: 98765432 ends in 32, which is a multiple of 8, and the number is divisible by 8 — but 132 ends in 32 and is not. Only the three-digit test settles it.
The pattern continues: 2^k needs the last k digits, because 10^k is divisible by 2^k. It is the same argument every time, which is why these three rules are really one.
7 and 13, and why they are awkward
Modulo 7, ten leaves remainder 3. Not 0, not 1, not −1 — so no tidy digit weighting exists, and there is no clean rule.
The standard trick is: chop off the last digit, double it, subtract it from the rest, and repeat. For 13 the trick is to multiply the last digit by 4 and ADD. These two are constantly swapped, and swapping them gives wrong answers on both.
Both are shown here with the working, and both are checked against the real remainder. For most numbers, honestly, dividing by 7 is faster than the trick — the rule is a curiosity rather than a shortcut, and it is worth saying so.
Combining rules, and when it fails
6 = 2 × 3, and 2 and 3 share no factor, so testing both is exactly the same as testing 6. The same works for 12 as 3 × 4, and for 15 as 3 × 5.
It fails when the two parts share a factor. Testing 12 as 2 × 6 does NOT work: 18 passes both — it is even and a multiple of 6 — and is not divisible by 12. The parts have to be coprime.
So split the divisor into prime powers: 12 as 4 × 3, not 2 × 6. 4 and 3 share nothing, and passing both really does settle it.
Sources and methodology
The rules are elementary modular arithmetic. These set out both the rules and the errors that circulate with them.
Method. Each rule is carried out on the digits exactly as it would be by hand, and the intermediate value is shown. Separately the number is divided by the divisor and the true remainder recorded. Both are displayed, and the verdicts must match — the test suite runs that comparison over two hundred thousand numbers. Any custom divisor is not given a rule at all; it is simply divided, because inventing a rule for an arbitrary divisor would be worse than doing the division. That engine is verified on every change against 64 hand-written assertions, including that every one of the thirteen rules agrees with the true remainder on every number up to two hundred thousand. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
FactorEvery factor of a number listed in the pairs that generate them, with the count derived from the prime exponents and separately checked against the list.
Prime FactorizationBreak any number into primes with the division ladder shown, the number of trial divisions reported, and the argument for why the search can stop at the square root.
Prime NumberWhether a number is prime, with a divisor named when it is not and the size of the search stated when it is. Deterministic, not probabilistic.
GCFThe greatest common factor of two to six numbers with all three routes shown — the shared primes to their lower powers, Euclid line by line, and the full factor lists when they are short enough to be honest.
LCMThe least common multiple of two to six numbers, with the prime table that produces it, the multiples themselves, and how far counting would have had to go.
Simplify FractionsReduce a fraction to lowest terms with both routes to the divisor shown side by side — Euclid line by line and the shared primes — and a stated proof when nothing can be cancelled.
These rules narrow a search; they do not replace one. If you want everything that divides a number, the Factor Calculator lists it, and the Prime Factorization Calculator gives the primes those factors are built from.
Educational use disclaimer
This is an educational tool. Every rule shown is also checked by dividing, so a rule stated wrongly would be caught rather than believed.
Published the divisibility page with every rule applied AND checked against the real remainder, because several of these rules circulate in wrong versions.
The rule for 8 is stated correctly as the last THREE digits; the two-digit version fails on 132. The rules for 7 and 13 are given with the right operations, since swapping subtract for add breaks both.
Nine of the thirteen rules are shown to be one idea — what remainder ten leaves modulo the divisor — rather than thirteen things to memorise.
Add this calculator to your site
Responsive embed — and private: nothing your visitors type leaves their browser.