Divisibility Rules for Competitive Exams

                                     Now I am going to discuss some basic divisibility conditions with you, which will help you in saving your time from unnecessary trying of natural numbers for division and utilising your time in other question.
In order to perform perfect and accurate division you will have to understand by heart these divisibility conditions and using them in your calculations for faster calculations.

Ø  Divisibility condition for 2 :- A number is divisible by 2 if the unit place digit is either 0 or an even number than the given number will be divisible by 2. Example 1230, 12344.

Ø  Divisibility condition for 3 :- A number is divisible by 3 if the sum of all the digits is divisible by 3, than the given number will be divisible by 3. Example 729, 12345.

Ø  Divisibility condition for 4 :- A number is divisible by 4 if the last two digits of the number is divisible by 4, than the given number will be divisible by 4. Example 1984, 12344.

Ø  Divisibility condition for 5 :- A number is divisible by 5 if its unit place digit is either 0 or 5, than the given number will be divisible by 5. Example 1235, 23470.

Ø  Divisibility condition for 6 :- A number is divisible by 6 if it fulfils the divisibility condition of 2 & 3, than the given number will be divisible by (2×3 = 6). Example 1296, 123426.

Ø  Divisibility condition for 7 :- A number is divisible by 7 if any digit is repeated six times in any number, than the given number will be  divisible by 7. Example 222222, 3333337.

Ø  Divisibility condition for 8 :- A number is divisible by 8 if the last three digits of any number are divisible by 8, than the given number will be divisible by 8. Example 125512, 1264840.

Ø  Divisibility condition for 9 :- A number is divisible by 9 if the sum of all the digits is divisible by 9, than the given number will be divisible by 9. Example 6561, 64845.

Ø  Divisibility condition for 10 :- A number is divisible by 10 if its unit place digit is  0, than the given number will be divisible by 10. Example 12350, 237430.

Ø  Divisibility condition for 11 :- A number is divisible by 11 if the difference between the sum of even places digits and odd places digits is 0 or a multiple of 11, than the given number is divisible by 11. Example 34683, 62271.

Explanation:-

Given number is 34683 in which sum of even places digits is (4+8 = 12) while the sum of odd places digits is (3+6+3 = 12). Now the difference between these two sum is (12-12 = 0) therefore the given number is divisible by 11. 

Given number is 62271 in which sum of even places digits is (2+7 = 9) while the sum of odd places digits is (6+2+1 = 9). Now the difference between these two sum is (9 - 9 = 0) therefore the given number is divisible by 11.

 

Divisibility condition for some special cases

Ø  Case 1^st :- (xⁿ + aⁿ) is exactly divisible by (x +a) if and only if when n is an odd number. Example (a⁵ + b⁵) is exactly divisible by (a+b).

Ø  Case 2^nd :- (xⁿ + aⁿ) is not exactly divisible by (x +a) if and only if when n is an even number. Example (a⁸ + b⁸) is not exactly divisible by (a+b).

Ø  Case 3^rd :- (xⁿ + aⁿ) is never exactly divisible by (x - a) either n is an odd number or an even number. Example (a⁷ + b⁷) or (a¹⁰ + b¹⁰)  is never exactly divisible by (a- b).

Ø  Case 4^th :- (xⁿ - aⁿ) is exactly divisible by (x +a) if and only if when n is an even number. Example (a¹² – b¹²) is exactly divisible by (a+b).

Ø  Case 5^th :- (xⁿ - aⁿ) is exactly divisible by (x - a) either n is an even number or an odd number. Example (a¹¹ - b¹¹) or (a¹⁸ - b¹⁸) is exactly divisible by (a- b).