The term divisibility is defined as the capacity of being divided. In mathematics, the property of being divided without any reminder is called divisibility. It is one of the important terms used in division functions in mathematics. Divisibility rules are defined as a set of general rules that are used to determine the nature of divisibility. The divisibility rules identify the nature of divisibility of a given number for another number. In other words, the divisibility rule is a short method of determining the result of the division function with or without a reminder.

Divisibility Rule

The term divisibility is a property of division function in mathematics. The term “divisible by” shows that a result is a whole number when a number is divided by another number. In mathematics, for all complex numbers as functions, a set of predefined rules predict the nature of divisibility for a number known as divisibility rules.

Different Rules on Divisibility

In mathematics, there are different rules to predict the nature of the division function and its result and reminder. These rules are described as follows.

For Divisor 1: 

  • Any number is a factor of 1. 
  • All the natural numbers are divisible by the number itself and by 1. 

For Divisor 2:

  • All the even numbers are divisible by 2. 
  • The number that has an even digit 0, 2, 4, 6, or 8 at a unit place is completely divisible by 2.

For Divisor 3:

  • If the sum of all the digits of a number is a multiple of 3, then the number is divisible completely by 3.

For example, the number is 753. The sum of its digits is 7 + 5 + 3 = 15. The sum of digits 15 is a multiple of 3. It shows that the number 753 is divisible by 3.

For Divisor 4:

  • The number in which unit place and tens place form a number that is multiple of 4 then the whole number is completely divisible by 4.
  • The digit at tens place is doubled, and multiplied by the digit at the unit place forms a number. If this number is multiple of 4, then the whole number is divisible by 4.

For example, the number is 65536. The double tens place digit is six, and it is multiplied by the unit place digit that is six × 6 = 36. 36 is a multiple of 4. It means the number 65536 is divisible by 4.

For Divisor 5:

  • All the numbers with 0 or 5 at their unit place are divisible by 5.

Divisibility Rule of 7

One of the important rules of indivisibility is the divisibility rule of 7.

  • Rule 1: For a given number, double the unit place digit and subtract it from the number formed by the rest of the digits; the resultant is a multiple of 7, then the given number satisfies the divisibility rule of 7.

Example: The number is 672. The double unit place digit is 4. After subtracting four from the number generated by the rest of the digits, 67 is 67-4 = 63. 63 is a multiple of 7; hence the number is divisible by 7 and satisfies the divisibility rule of 7.

  • Rule 2: For a given number, the unit place digit is multiplied by five and add it to the number formed by the rest of the digits; the resultant is multiple of 7, then the given number is satisfied by the divisibility rule of 7.

Divisibility Rule of 11

Another important rule of indivisibility is the divisibility rule of 11.

  • Rule 1: If the difference of the numbers formed by alternating digits in a number is a multiple of 11, then that number satisfies the divisibility rule of 11. 

Example: The number is 2143. The numbers formed by the alternate digits are 24 and 13. The difference of the numbers 24 -13 = 11. 11 is one of the factors of 11. It means the number 2143 satisfies the divisibility rule of 11.

  • Rule 2: Subtract the unit place digit of the number from the rest of the number. If the resultant value is a multiple of 11, then the original number will satisfy the divisibility rule of 11.

Example: The number is 957. The number of unit place digit seven is subtracted from the rest; the resultant is 95-7 = 88. The resultant is a multiple of 11. Hence the number 957 satisfies the divisibility rule of 11.  

Facts About Divisibility

There are a few facts on divisibility in mathematics which are given below.

  • Every number is divisible by 1. 
  • If a number is completely divisible by another number (divisor), then it will be divisible by all the divisor factors.
  • The result of divisibility is always the complete number.

Conclusion

The division is a universal operation. The division is also known as the inverse operation of multiplication in mathematics. To make this division operation simpler, there are different rules given, which are rules of divisibility. These rules are predefined and tested on different numbers. There is a different rule for the different divisors. In mathematics, not all the numbers are completely divisible by the other numbers, so the divisibility rules are invented. The divisibility rules for the divisors are the shortcut methods of determining a number’s actual divisor by examining the digits that make the number.