Question

What are the divisibility rules and how many are there?

Answer 1

Hint: In this question, we have been asked to define divisibility rules and we have been asked how many divisibility rules are there. Along with that, also define the divisibility rules of some numbers.

Complete step-by-step solution:
Let us define divisibility rules first.
Divisibility rules – These rules are the rules which help in determining whether a given number is divisible by the given divisor or not, without actually performing the division and just by looking for certain simple tricks and tests.
There are many divisibility rules but some of the most common divisibility rules which are used quite often are as follows:
$ \Rightarrow $Divisibility rule of $2$ – The divisibility rule of $2$ says that if the ones digit of the given number is $0$, $2$, $4$, $6$ or $8$, then the number is divisible by $2$. Basically, the number should be an even number.
$ \Rightarrow $Divisibility rule of $3$ – The divisibility rule of $3$ says that if the sum of all the digits of a number is divisible by $3$, then the entire number is divisible by $3$.
$ \Rightarrow $Divisibility rule of $4$ – The divisibility rule of $4$ states that if the last two digits of a number is divisible by $4$, then the entire number is divisible by $4$. It does not matter how many digits a number has. This rule will always apply.
$ \Rightarrow $Divisibility rule of $5$ – The divisibility rule of $5$ says that if the last digit of a number is either $0$ or $5$, then the number is divisible $5$.
$ \Rightarrow $Divisibility rule of $6$ – The divisibility rule of $6$ says that if the number is divisible by $2$ and $3$ both then, the number is divisible by $6$. So, for this number you will have to check the divisibility by $2$ and $3$ both.
$ \Rightarrow $Divisibility rule of $7$– The divisibility rule of $7$ says that remove the last digit of the number, double it and then subtract it from the remaining digits. If the final answer is $0$ or a multiple of $7$, then the number is divisible by $7$.
$ \Rightarrow $Divisibility rule of $8$– The divisibility rule of $8$ says that if the last three digits of a number is divisible by $8$, then the entire number is divisible by $8$. It does not matter how many digits a number has. This rule will always apply.
$ \Rightarrow $ Divisibility rule of $9$– The divisibility rule of $9$ says that if the sum of all the digits of a number is divisible by $9$, then the entire number is divisible by $9$.
$ \Rightarrow $ Divisibility rule of $10$– The divisibility rule of $10$ says that if the last digit of a number is $0$ then the number is divisible $10$.
$ \Rightarrow $ Divisibility rule of $11$- The divisibility rule of $11$ says that first, add the digits at odd and even numbers separately. Then, subtract them. If the final answer is $0$ or a multiple of $11$, then the number is divisible by $11$.

Note: There are many other numbers that have their own divisibility rules but these are the most common rules and are used most frequently. If you have a big divisor, break it into its prime factors and check the divisibility. If the dividend is divisible by all the prime factors of the divisor, then the number is divisible.