What this calculator does
Divisibility rules let you test a number without doing the division. Most rest on properties of the digits: the digit sum settles 3 and 9, the last digit settles 2, 5 and 10, and the last two or three digits settle 4 and 8.
Some numbers are far more divisible than their size suggests. 360 divides evenly by nine of the numbers from 2 to 12, failing only on 7 and 11, which is exactly why it was chosen for the degrees in a circle.
The formula
Each divisor from 2 to 12 is tested by taking the remainder. The digit sum is also reported, since it is the basis of the rules for 3 and 9.
| Term | Meaning |
|---|---|
| Divisibility | Dividing exactly with no remainder. |
| Digit sum | The sum of the digits, which determines divisibility by 3 and 9. |
| Highly composite | A number with unusually many divisors for its size, such as 360. |
The inputs explained
| Field | What to enter |
|---|---|
| Whole number | The whole number to test. The sign is ignored. |
When to use it
Simplifying a fraction
Finding common divisors is the first step in reducing a fraction.
Checking a factorisation
Knowing the small divisors narrows the search quickly.
Learning the rules
Seeing which divisors succeed alongside the digit sum makes the rules concrete.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How divisible is each number?
Divisors from 2 to 12 tested on each.
Questions
Why does the digit sum work for 3 and 9?
Because every power of ten leaves a remainder of 1 when divided by 9. That means a number and its digit sum always have the same remainder modulo 9, and since 3 divides 9, the same trick works for 3.
What is the rule for 7?
There is one, but it is awkward: double the last digit and subtract it from the rest, repeating until the result is recognisable. For most purposes dividing directly is faster, which is why 7 has a reputation as the difficult case.
What is the rule for 11?
Alternately add and subtract the digits. If the result is divisible by 11, so is the original. For 1001 that gives 1 − 0 + 0 − 1 = 0, which is divisible by 11, and indeed it is.
Why is 360 so useful?
Because it has 24 divisors, far more than its neighbours. That divisibility is why it was adopted for degrees in a circle: it splits evenly into halves, thirds, quarters, fifths, sixths, eighths, ninths, tenths and twelfths.
For reducing the digits to one, see the digital root calculator. For consecutive integer sums, see the consecutive integers calculator.