### Theory:

Divisibility by $$5$$: If a digit in the ones place of a number is $$5$$ or $$0$$, then it is divisible by $$5$$.
Example:
1. Let us take the numbers $$95$$ and $$680$$.

Rule for $$5$$: Unit digit of a number is either $$5$$ or $$0$$.

The numbers $$95$$ and $$680$$ ends in $$5$$ and $$0$$.

Therefore, $$95$$ and $$680$$ are divisible by $$5$$.

2. Let us take the numbers $$462$$ and $$327$$.

Rule for $$5$$: Unit digit of a number is either $$5$$ or $$0$$.

The numbers $$462$$ and $$327$$ ends in $$2$$ and $$7$$.

Therefore, $$462$$ and $$327$$ are not divisible by $$5$$.
Divisibility by $$6$$: If a number is divisible by $$2$$ and $$3$$, then that number is divisible by $$6$$.
Example:
1. Let us take the number $$246$$.

Rule for $$6$$: Number is divisible by $$2$$ and $$3$$.

It is divisible by $$2$$ as it ends with $$6$$.

Now, $$2+4+6=12$$. $$12$$ is divisible by $$3$$, so $$246$$ is divisible by $$3$$ aslo.

This shows that $$246$$ is divisible by $$2$$ and $$3$$.

Therefore, $$246$$ is divisible by $$6$$.

2. Let us take the number $$154$$.

Rule for $$6$$: Number is divisible by $$2$$ and $$3$$.

It is divisible by $$2$$ as it ends with $$4$$.

Now, $$1+5+4=10$$. $$10$$ is not divisible by $$3$$.

Thus, $$154$$ is not divisible by $$3$$.

Therefore, $$154$$ is not divisible by $$6$$.
Divisibility by $$7$$: Double the last digit of the number and then subtract it from the remaining number if the number formed is divisible by $$7$$, then the number is divisible by $$7$$.
Example:
1. Let us take the number $$2548$$.

Rule for $$7$$: Double the last digit and subtract it from the remaining number and check the result is divisible by $$7$$.

Doubling the last digit $$8$$ results as $$2\times 8 =16$$.

Subtract it from the remaining number $$= 254-16 = 238$$.

Now the number $$238\div 7 = 34$$.

Therefore, $$2548$$ is divisible by $$7$$.

2. Let us take the number $$810$$.

Rule for $$7$$: Double the last digit and subtract it from the remaining number and check the result is divisible by $$7$$.

Doubling the last digit $$0) results as \(2\times 0 = 0$$.

Subtract it from the remaining number $$= 81- 0 = 81$$.

Now the number $$81\div 7$$ will leave the remainder $$4$$.

Therefore, $$810$$ is not divisible by $$7$$.
Divisibility by $$8$$: A number is divisible by $$8$$ if the number formed by its last three digits is divisible by $$8$$.
Example:
1. Let us take the number $$2544$$.

Rule for $$8$$: Last three digits is divisible by $$8$$, then the number is divisible by $$8$$.

The last $$3$$ digits are $$544$$ and divide it by $$8$$.

$$544\div8 = 68$$.

Therefore, $$2544$$ is divisible by $$8$$.

2. Let us take a number $$1260$$.

Rule for $$8$$: Last three digits is divisible by $$8$$, then the number is divisible by $$8$$.

The last $$3$$ digits are $$260$$ and divide it by $$8$$.

$$260\div8$$ leaves a remainder $$4$$.

Therefore, $$1260$$ is not divisible by $$8$$.