Divisibility rules calculator

What is a divisibility rules calculator?

A divisibility rules calculator takes a single whole number $N$ and tells you, at a glance, which small divisors it is divisible by. This tool checks every divisor from 2 to 12 and answers Yes or No for each one. A number is divisible by $d$ when the division leaves no remainder — that is, when $N$ can be written as $d$ multiplied by another whole number.

Divisibility is one of the most useful ideas in arithmetic. It lets you simplify fractions, find common factors, spot prime numbers, and split quantities into equal groups without doing long division every time.

What does “divisible” mean?

We say $N$ is divisible by $d$ when the remainder of the division is zero:

$$N \bmod d = 0$$

Equivalently, there is a whole number $k$ such that:

$$N = d \times k$$

For example, 60 is divisible by 12 because $60 = 12 \times 5$, leaving no remainder. In contrast, 60 is not divisible by 7, because $60 = 7 \times 8 + 4$ — a remainder of 4 is left over.

Because $0 \bmod d = 0$ for every non-zero $d$, the number 0 is divisible by every divisor. The calculator treats 0 as a valid input and answers Yes across the whole grid.

How does the calculator work?

You type an integer, and the calculator applies a known divisibility rule for each divisor from 2 to 12. Each rule is a shortcut that avoids full division by looking only at the digits of the number.

Divisibility by 2

A number is divisible by 2 when it is even — its last digit is 0, 2, 4, 6, or 8. Example: 60 ends in 0, so it is divisible by 2.

Divisibility by 3

A number is divisible by 3 when the sum of its digits is divisible by 3. For 60: $6 + 0 = 6$, and 6 is divisible by 3, so 60 is divisible by 3.

Divisibility by 4

A number is divisible by 4 when the number formed by its last two digits is divisible by 4. For 60, the last two digits form 60, and $60 = 4 \times 15$, so 60 is divisible by 4.

Divisibility by 5

A number is divisible by 5 when its last digit is 0 or 5. Example: 60 ends in 0, so it is divisible by 5.

Divisibility by 6

A number is divisible by 6 when it is divisible by both 2 and 3. Because 60 passes the tests for 2 and 3, it is divisible by 6.

Divisibility by 7

Take the last digit, double it, and subtract it from the rest of the number. If the result is divisible by 7, so is the original. For 91: remove the 1, double it to get 2, then $9 - 2 = 7$. Since 7 is divisible by 7, so is 91 — indeed $91 = 7 \times 13$.

Divisibility by 8

A number is divisible by 8 when the number formed by its last three digits is divisible by 8. For numbers with fewer than three digits, simply check the whole number.

Divisibility by 9

A number is divisible by 9 when the sum of its digits is divisible by 9. For 60: $6 + 0 = 6$, which is not divisible by 9, so 60 is not divisible by 9.

Divisibility by 10

A number is divisible by 10 when its last digit is 0. Example: 60 ends in 0, so it is divisible by 10.

Divisibility by 11

Add and subtract the digits in an alternating pattern (from right to left) and check whether the result is divisible by 11 (zero counts as divisible). For 121: $1 - 2 + 1 = 0$, so 121 is divisible by 11.

Divisibility by 12

A number is divisible by 12 when it is divisible by both 3 and 4. Because 60 passes the tests for 3 and 4, it is divisible by 12.

Formula

The general test for any divisor $d$ relies on the modulo (remainder) operation:

$$N \text{ is divisible by } d \iff N \bmod d = 0$$

Many rules build on the fact that a number is congruent to a simple combination of its digits. For the digit-sum rules (3 and 9), if $N$ has digits $a_i$, then:

$$N \equiv \sum_i a_i \pmod{9}$$

so $N$ is divisible by 3 or 9 exactly when its digit sum is. For 11, the alternating sum applies because $10 \equiv -1 \pmod{11}$:

$$N \equiv \sum_i (-1)^i a_i \pmod{11}$$

Worked examples

Example 1: N = 60

• Divisible by 2, 3, 4, 5, 6, 10, and 12 → Yes
• Divisible by 7, 8, 9, and 11 → No

The number 60 is highly composite, which is exactly why calendars, clocks, and measurement systems favour it: it splits evenly into halves, thirds, quarters, fifths, sixths, and twelfths.

Example 2: N = 91

• Divisible by 7 → Yes (because $91 = 7 \times 13$)
• Divisible by 2 → No (91 is odd)
• Divisible by 3 → No (digit sum $9 + 1 = 10$ is not divisible by 3)

Example 3: N = 0

Zero is divisible by every divisor, since $0 \bmod d = 0$ for each $d$. The calculator answers Yes for all of 2 through 12.

Practical notes

• The rules apply to whole numbers. Negative numbers follow the same rules as their absolute values — for instance, −60 behaves like 60.
• Divisibility rules are the fastest way to reduce a fraction to lowest terms: find a divisor that divides both the numerator and denominator, then cancel it.
• To find every divisor of a number rather than just checking 2 through 12, use the Factor calculator.
• To combine several numbers, the Least common multiple calculator and the Greatest common factor calculator build directly on divisibility.

Frequently asked questions

Is every number divisible by 1?

Yes. Every whole number is divisible by 1 and by itself, which is why the divisibility grid starts at 2 — the first non-trivial divisor.

Why is 0 divisible by every number?

Because $0 \bmod d = 0$ for every non-zero divisor $d$: dividing 0 by any number leaves no remainder. Note that division by 0 is undefined, which is why 0 never appears as a divisor.

Is there a single rule for divisibility by 6 or 12?

Not a digit trick of its own — you combine simpler rules. A number is divisible by 6 when it passes both the 2 and 3 tests, and divisible by 12 when it passes both the 3 and 4 tests.