Divisibility criteria

We know how to find the divisors of a given number, by dividing it by different candidate numbers.

Nevertheless, there are some simple rules that allow us, at first sight, to deduce some divisors.

A number is divisible by $2$ if it ends with a $0$ or an even number.

Example

$44, 56, 238, 70, 92, 122$ .

A number is divisible by $3$ if the sum of its digits is $3$ or a multiple of $3$ .

Example

$363, 54, 81, 111, 1.320, 207$ .

A number is divisible by $4$ if its last two digits are zeros or multiples of $4$ .

Example

$408, 300, 1.216, 312, 43.332, 5.000$

A number is divisible by $5$ if it ends with a $0$ or a $5$ .

Example

$45, 500, 134.325, 34.200, 665, 10$ .

A number is divisible by $6$ if it is divisible by $2$ and also by $3$ .

Example

$3.030, 4.410, 36, 12, 132, 66$ .

A number is divisible by $7$ if the difference of the number without the digit of the units and the double of the digit of the units is $0$ or a multiple of $7$ .

Example

$126$ is divisible by $7$ because: $12 - (6 \times 2) = 12 - 12 = 0$

$224$ is divisible by $7$ because: $22 - (4 \times 2) = 22 - 8 = 14$ , that is a multiple of $7$ .

$567$ is divisible by $7$ because: $56 - (7 \times 2) = 56 - 14 = 42$ , that is a multiple of $7$ .

A number is divisible by $9$ if the sum of its digits gives a multiple of $9$ .

Example

$333, 999, 810, 945, 360, 9.963$

A number is divisible by $10$ if the digit of the units is $0$ .

Example

$20, 43.340, 620, 34.230, 100.000, 440$

A number is divisible by $11$ if the difference of the sum of the digits that are in even places and in odd places is $0$ or a multiple of $11$ .

Example

$242$ is divisible by $11$ because: $(2 + 2) - 4 = 4 - 4 = 0$

$616$ is divisible by $11$ because: $(6 + 6) - 1 = 12 - 1 = 11$

$96.954$ is divisible by $11$ because: $(9 + 9 + 4) - (6 + 5) = 22 - 11 = 11$

A number is divisible by $25$ if its last two digits are zeros or a multiple of $25$ .

Example

$3.300, 1.250, 375, 25.425, 100, 25.050$

A number is divisible by $125$ if its last three digits are zeros or a multiple of $125$ .

Example

$20.000, 1.250, 34.125, 375, 501.125, 1.000$