What are Divisibilty tests/rules?

Divisibility tests or rules can be explained as shortcut methods that are used to determine if a given number is completely divisible by another number i.e. the remainder is zero.
For example to determine if a number is divisible by 2, one need not actually do the complete division. But if the given number ends with 2,4,6,8 or 0, one can say with certainty that it is divisible by 2.

Divisibilty tests/rules

A positive integer $N$ is divisible by $2$ if the last digit of $N$ is $2, 4, 6, 8$ or $0$

For example:
a) Is 176 divisible by 2?
Solution
The last digit of 176 is 6.
∴ 176 is divisible by 2. A positive integer $N$ is divisible by $3$ if the sum of the digits of $N$ is a multiple of $3$

For example:
a) Is 1131 divisible by 3?
Solution
The sum of digits of 1131 i.e. $1 + 1 + 3 + 1 = 6$ is a multiple of 3
∴ 1131 is divisible by 3. A positive integer $N$ is divisible by $4$ if the last two digits of $N$ are a multiple of $4$ or if the last 2 digits are $00$

For example:
a) Is 1264 divisible by 4?
Solution
The number formed by the last 2 digits of 1264 i.e. 64 is divisible by 4
∴ 1264 is divisible by 4.

b) Is 2600 divisible by 4?
Solution
The last 2 digits of 2600 are 00
∴ 2600 is divisible by 4. A positive integer $N$ is divisible by $5$ if the last digit of $N$ is either a $0$ or a $5$

For example:
a) Is 135 divisible by 5?
Solution
The last digit of 135 is 5
∴ 135 is divisible by 5. A positive integer $N$ is divisible by $6$ if it is divisible by both $2$ and $3$

For example:
a) Is 96 divisible by 6?
Solution
96 is divisible by 2 as the last digit is 6
96 is also divisible by 3 as the sum of the digits i.e. $9 + 6 = 15$ is a multiple of 3
∴ 96 is divisible by 6. A positive integer $N$ is divisible by $7$ if subtracting twice the last digit of $N$ from the remaining digits gives a $0$ or a multiple of $7$

For example:
a) Is 168 divisible by 7?
Solution
Twice the last digit (i.e. 8) is 16.
Subtract 16 from the number formed by the remaining digits i.e. 16
⇒ 16 - 16 = 0
∴ 168 is divisible by 7.

b) Is 441 divisible by 7?
Solution
Twice the last digit (i.e. 1) is 2.
Subtract 2 from the number formed by the remaining digits i.e. 44
⇒ 44 - 2 = 42
∴ As 42 is a multiple 7, 441 is divisible by 7. A positive integer $N$ is divisible by $8$ if the last three digits of $N$ are a multiple of $8$

For example:
a) Is 1864 divisible by 8?
Solution
The number formed by the last 3 digits of 1864 i.e. 864 is divisible by 8
∴ 1864 is divisible by 8.

b) Is 14032 divisible by 8?
Solution
The number formed by the last 3 digits of 14032 i.e. 032 is divisible by 8
∴ 14032 is divisible by 8. A positive integer $N$ is divisible by $9$ if the sum of the digits of $N$ is a multiple of $9$

For example:
a) Is 1134 divisible by 9?
Solution
The sum of digits of 1134 i.e. $1 + 1 + 3 + 4 = 9$ is a multiple of 9
∴ 1134 is divisible by 9. A positive integer $N$ is divisible by $10$ if the last digit of $N$ is $0$

For example:
a) Is 3440 divisible by 10?
Solution
The last digit of 3440 is 0.
∴ 3440 is divisible by 10.