**Rules of Divisibility**

**2:**A number is

**divisible by 2**if the last digit (unit's digit) is even. e.g. 3

**2**, 45999

**2**

**3:**A number is

**divisible by 3**, if the sum of digits of a number is divisible by 3

**.**

e.g. 252 = 2 + 5 + 2 = 9 ÷ 3 = 3 ∴ 252 is divisible by 3

**4:**A number is

**divisible by 4**, if the last two digits of the number is divisible by 4.

e.g. 819

**24**= since last two digits 24 is divisible by 4, hence the number.

**5:**A number is

**divisible by 5**, if the last digit of the number is 0 or 5.

e.g. 3

**5**, 20

**0**, 100

**5**

**6:**A number is

**divisible by 6**, if the number is divisible by both 2 and 3.

**7:**A number is

**divisible by 7**, to check for this follow these steps:

- Take the last digit (right most) and double it.
- Subtract the doubled number from the remaining number.
- Check the answer if it divisible by 7 and Repeat the above two steps.

= 472 - (2 × 5) = 472 - 10 = 462

= 46 - (2 × 2 ) = 46 - 4 = 42

Since 42 is divisible by 7. ∴ number 47292 is divisible by 7.

**8:**A four digit or higher number is divisible by 8, if its last 3 digits are divisible by 8.

e.g. 34064 Since last three digits forms 064 which is divisible by 8.

**9:**A number is

**divisible by 9**, if sum of digits of the number is divisible by 9.

**10:**A number is

**divisible by 10**, if the last digit is 0, then 10 divides that number.

**11:**A number is

**divisible by 11**, to check this take the sum of all the digits at odd places and the sum of digits at even places. If the difference of these two sums is zero or multiple of 11, then the number is divisible by 11.

e.g. 8679 = (8 + 7) - (6+9) = 15 - 15 = 0.

e.g. 246906 = (2+6+0) - (4+ 9 + 6) = |8 - 19| = 11.

**12:**A number is

**divisible by 12**, if the number is divisible by 4 and 3 both.

**13:**To check if a number is

**divisible by 13**, take four times of the last digit and add it to the remaining number. Repeat the step if necessary. If the result number is multiple of 13, the number is divisible by 13.

e.g. 3172 = 317 + (4×2) = 317 + 8 = 325 = 32 + (4 × 5) = 32 + 20 = 52 which multiple of 13.

**19:**To check if a number is

**divisible by 19**, take the double of last digit and add it to remaining number. Repeat the steps. If the result is divisible by 19, hence the original number.

e.g. 1292 = 129 + (2 × 2) = 133 = 13 + (2 × 3) = 19. ∴ number 1292 is divisible by 19

**24:**A number is

**divisible by 24**, if it is divisible by 3 and 8 both.

**25:**A number is

**divisible by 25**, if the last two digits are divisible by 25.

**125:**A number is

**divisible by 125**, if the last three digits are divisible by 125.

**Q1: Check if 312 is divisible by 13 and 19.**

Answer: 312 = 31 + (4×2) = 39 is divisible by 13

**Q2: Check if 1329 and 2113 are divisible by 9**

Answer: 1629 = 1 + 6 + 2 + 9 = 18, number is divisible by 9.

2113 = 2 + 1 + 1 + 3 = 7, number is not divisible by 9.

**Q3: The number 477 is divisible by the following numbers except**

(a) 3

(b) 6

(c) 9

(d) None of these

Answer: (b) 6

**Q4: 6721 is divisible by**

(a) 3

(b) 7

(c) 8

(d) 11

Answer: (d) 11

6721 = 6+7+2+1 = 16, not divisible by 3

6721 = 672 - (2×1) = 670 = 67 - 0 = 67 which is not divisible by 7

6721 = last three digits 721 not divisible by 8

6721 = (6+2) - (7+1) = 8 - 8 = 0, number is divisible by 11.

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