### Theory:

When a given number is divisible by another number without leaving a remainder, then the given number is said to be divisible of another number.
Divisibility rule gives a precise method to determine whether a given integer is divisible by a fixed divisor.
In this, we shall see the various types of divisibility. They are as follows:
• Divisibility by $$2$$.
• Divisibility by $$3$$.
• Divisibility by $$4$$.
• Divisibility by $$5$$.
• Divisibility by $$6$$.
• Divisibility by $$8$$.
• Divisibility by $$9$$.
• Divisibility by $$11$$.
Divisibility by $$2$$: If the number ends at $$2$$, $$4$$, $$6$$, $$8$$ or $$0$$, it is divisible by $$2$$.
Example:
1. Let us take the numbers $$28$$, $$54$$, $$96$$.

Rule for $$2$$: Number ends at $$2$$, $$4$$, $$6$$, $$8$$ or $$0$$.

Here $$28$$, $$54$$, and $$96$$ ends with $$8$$, $$4$$, and $$6$$ respectively.

Hence, $$28$$, $$54$$ and $$96$$ are divisible by $$2$$.

2. Let us take the numbers $$35$$, $$57$$, $$1297$$.

Rule for $$2$$: Number ends at $$2$$, $$4$$, $$6$$, $$8$$ or $$0$$.

Here $$35$$, $$57$$, $$1291$$ ends with $$5$$, $$7$$, and $$1$$ respectively.

Hence, $$28$$, $$54$$ and $$96$$ are not divisible by $$2$$.
Divisibility by $$3$$: If the sum of its digits of any number is divisible by $$3$$ then that number is divisible by $$3$$.
Example:
1. Let us take the number $$429$$.

Rule for $$3$$: Sum of the digits of the number is divisible by $$3$$.

$$4+2+9=15$$; $$15\div3=5$$

Hence, $$429$$ is divisible by $$3$$.

2. Let us take the number $$512$$.

Rule for $$3$$: Sum of the digits of the number is divisible by $$3$$.

$$5+1+2$$ $$=$$ $$8\div3$$. This division leaves a remainder $$2$$.

Hence, $$512$$ is not divisible by $$3$$.
Divisibility by $$4$$: If a last two digits of any number are divisible by $$4$$, then that number is divisible by $$4$$.
Example:
1. Let us look at the number $$628$$.

Rule for $$4$$: Last $$2$$ digits of the number is divisible by $$4$$.

Last $$2$$ digits are $$28$$ and $$28\div4=7$$.

Hence, $$628$$ is divisible by $$4$$.

2. Let us look at the number $$714$$.

Rule for $$4$$: Last $$2$$ digits of the number is divisible by $$4$$.

Last $$2$$ digits are $$14$$ and $$14\div4$$. This division leaves a remainder $$2$$.

Hence, $$714$$ is not divisible by $$4$$.