Divisibility

Divisibility:
If a number divides another number without leaving remainder then second is said to be divisible by first.
Following are tips to check divisibility of numbers:


1. Divisibility by 2:A number is divisible by 2 if its unit digit is 0,2,4,6 or 8.


Example: 64578 is divisible by 2 or not?
Step 1 - Unit digit is 8.

Result: 64578 is divisible by 2.


Example: 64575 is divisible by 2 or not?
Step 1 - Unit digit is 5.

Result: 64575 is not divisible by 2.


2. Divisibility by 3: A number is divisible by 3 if sum of its digits is completely divisible by 3.
Example: 64578 is divisible by 3 or not?
Step 1 - Sum of its digits is 6 + 4 + 5 + 7 + 8 = 3. which is divisible by 3.
Result: 64578 is divisible by 3.
Example: 64576 is divisible by 3 or not?
Step 1 - Sum of its digits is 6 + 4 + 5 + 7 + 6 = 2. which is not divisible by 3.
Result: 64576 is not divisible by 3.


3. Divisibility by 4: A number is divisible by 4 if number formed using its last two digits is completely divisible by 4.
Example: 64578 is divisible by 4 or not?
Step 1 - number formed using its last two digits is 7. which is not divisible by 4.
Result: 64578 is not divisible by 4.
Example: 64580 is divisible by 4 or not?
Step 1 - number formed using its last two digits is 8. which is divisible by 4.
Result: 64580 is divisible by 4.


4. Divisibility by 5: A number is divisible by 5 if its unit digit is 0 or 5.
Example: 64578 is divisible by 5 or not?
Step 1 - Unit digit is 8.
Result: 64578 is not divisible by 5.
Example: 64575 is divisible by 5 or not?
Step 1 - Unit digit is 5.
Result: 64575 is divisible by 5.


5. Divisibility by 6: A number is divisible by 6 if the number is divisible by both 2 and 3.
Example: 64578 is divisible by 6 or not?
Step 1 - Unit digit is 8. Number is divisible by 2.
Step 2 - Sum of its digits is 6 + 4 + 5 + 7 + 8 = 3. which is divisible by 3.
Result: 64578 is divisible by 6.


Example: 64578 is divisible by 6 or not?

Step 1 - Unit digit is 6. So, Number is divisible by 2.

Step 2 - Sum of its digits is 6 + 4 + 5 + 7 + 6 = 2. which is not divisible by 3.
Result: 64576 is not divisible by 6.

6. Divisibility by 7:
A number of 2 digits is divisible by 7 because 3 x 6 + 3 = 21. 21 is divisible by 7.
A number of three or more digits is divisible by 7 if the sum of the numbers formed by the last two digits and twice the number formed by the remaining digits is divisible by 7.
For Example:
(i) 574 is divisible by 7 because 74 + 5 x 2 = 74 + 10 = 84 is divisible by 7.
(ii) 2268 is divisible by 7 because 68 + 22 x 2 = 68 + 44 = 112 is divisible by 7.

7. Divisibility by 8: A number is divisible by 8 if number formed using its last three digits is completely divisible by 8.
Example: 64578 is divisible by 8 or not?
Step 1 - number formed using its last three digits is 57. which is not divisible by 8.
Result: 64578 is not divisible by 8.
Example: 64576 is divisible by 8 or not?
Step 1 - number formed using its last three digits is 57. which is divisible by 8.
Result: 64576 is divisible by 8.

8. Divisibility by 9: A number is divisible by 9 if sum of its digits is completely divisible by 9.
Example: 64579 is divisible by 9 or not?
Step 1 - Sum of its digits is 6 + 4 + 5 + 7 + 9 = 3. which is not divisible by 9.
Result: 64579 is not divisible by 9.
Example: 64575 is divisible by 9 or not?
Step 1 - Sum of its digits is 6 + 4 + 5 + 7 + 5 = 2. which is divisible by 9.
Result: 64575 is divisible by 9.

9. Divisibility by 10: A number is divisible by 10 if its unit digit is 0.
Example: 64575 is divisible by 10 or not?
Step 1 - Unit digit is 5.
Result: 64578 is not divisible by 10.
Example: 64570 is divisible by 10 or not?
Step 1 - Unit digit is 0.
Result: 64570 is divisible by 10.

10. Divisibility by 11: A number is divisible by 11 if difference between sum of digits at odd places and sum of digits at even places is either 0 or is divisible by 11.
Example: 64575 is divisible by 11 or not?
Step 1.Difference between sum of digits at odd places
and sum of digits at even places = (6+5+5) - (4+7) = 5
which is not divisible by 11.


Result: 64575 is not divisible by 11.

Example: 64075 is divisible by 11 or not?
Step 1.Difference between sum of digits at odd places
and sum of digits at even places = (6+0+5) - (4+7) = 0.
Result: 64075 is divisible by 11.

Tips on Division: 1. If a number n is divisible by two co-primes numbers a, b then n is also divisible by ab.
2. (a-b) always divides (an - bn) if n is a natural number.
3. (a+b) always divides (an - bn) if n is an even number.
4. (a+b) always divides (an + bn) if n is an ODD Number.


Division Algorithm:
When a number is divided by another number then
Dividend = (Divisor x Quotient) + Reminder