RULES OF DIVISIBILITY
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RULES OF DIVISIBILITY
1.
Divisibility by 2 :
If the unit’s place digit is zero or divisible by 2; then the number
will be divisible by 2.
Example: 248, 3782, 47986 In all these numbers the last digit is divisible by 2, therefore all the numbers are divisible by 2.( check by actual division)
2.
Divisibility by 3:
If the sum of all digits of the number is divisible by 3 the number
divisible by 3.
Let us check the rule:
1191 ( sum of digits 1+1+9+1=12 which is divisible by 3) so the number is also divisible by three.
4645 ( sum of digits 4+6+4+5=19 which is not divisible by 3) so the number is also not divisible by 3
Check by performing actual division.
3.
Divisibility by 4:
If any number has 00 at unit’s and ten’s place or the number formed by
the digits of ten’s place and unit’s place is divisible is divisible by4 then
the number is divisible by 4.
Example:
4624 ( last two digit is 24 which is divisible by 4, so the number is also divisible by 4)
8122 ( last two digits 22 is not divisible by 3, so the number is also not divisible by 4)
4.
Divisibility by 5:
If the unit’s place digit is 0 or
5 then the number is divisible by 5.
Example:
4500, 5535, 7935, and 1245 are divisible by 5 because the unit place digit is either 0 or 5
5. Divisibility by 6: If the number is divisible by 2 and 3 separately then the number is divisible by 6.
Example :
(25134 is divisible by 2 as
it contains 4 as unit’s place digit and the sum of all the digits is divisible
by 3 hence 25134 will be divisible by 6.)
6.
Divisibility by 7:
(i)
When a number is of two digits: Add
the three times of ten’s place digit to its unit’s place digit.If the resultant
is divisible by 7; the number is divisible by 7. This rule is applicable for 3
digits number also.
(ii)
When the number is of made up of 3 or
4 digits: Double the number made of the digits at 100’th and 1000’th place and
it to the number made of unit’s and 10’th place digits. If this sum is
divisible by 7 then the number is divisible by 7.
(iii)
If the number consists of more than 4
digits: while deciding the divisibility of any number containing more than 4
digits we proceed as follows ---
(a) Make groups of 3 digits each from right to left
(b) Now substract 2nd set
from 1st set and add 3rd set and substract 4th
set. This process is repeated till all the digits are taken into consideration.
(c) If the result obtained is either zero or divisible by 7 then the number will be divisible by 7. If the number obtained above is a two digits number, its divisibility can be tested orally.
Example :
( 43752786 = 43/752/786 now 786 – 752 + 43 = 77 which is divisibility by 7.)
(iv)
One more general rule of
divisibility( For any number): Double the unit’s place digit and subtract it
from the remaining digits. Continue till we get a number of two digit. If it is
divisible by 7 then the number is divisible by 7.
6.
Divisibility by 8:
If the number formed by last
three digits of a given number is divisible by 8 or if there are three zeros in
the end then the given number is divisible by 8.
Example: 4000 ( last three digits are zero therefore divisible by 8
1936 ( last three digits are 936 which is divisible by 8 therefore the given number is divisible by 8 )
7.
Divisibility by 9:
If the sum of all the digits of
any number is divisible by 9 then the number is divisible by 9.
Example: 2178 ( sum of digits = 2+1+7+8 = 18 which is divisible by 9 therefore the given number is also divisible by 9)
8.
Divisibility by 10 : Any number ending with 0 is divisible by 10.
Since 0 at unit place satisfies the condition of divisibility by 2 and 5 hence
the number divisible by 10 is also divisible by 2 and 5.
Examples: 1240, 3980, 4900, 12340 are divisible by 10 because all the numbers are ending with zero.)
9.
Divisibility by 11 : If he
difference between the sum of the digits in odd places and the sum of the
digits at even places is either zero or a number divisible by 11 then the
number being tested is a multiple of 11, otherwise our number is not exactly
divisible by 11.
10. Divisibility by 12 : If any number is separately divisible by 3 and
4 then the number is divisible by 12.
11. Divisibility by 13 : In this process we employ the (c) method of divisibility by 7.
(check the divisibility of 26132132 by
13, making sets of three 26/132/132 Now
132 – 132 +26 = 26 which is divisible by 13 , so the number is divisible by
13.)
If any number contains 2 or 3 digits
then add 4 times of unit’s place digits to the remaining number and if the sum
of this process is divisible by 13 then the number is divisible by 13.
12. Divisibility by 14 : If any
number is separately divisible by 2 and 7 then the given number is divisible by 14.
13. Divisibility by 15: If any number satisfies the
divisibility condition of 3 and 5 separately the number is divisible by 15.
14. Divisibility by 16 : In a given number if the number
formed by last 4 digits starting from unit’s place is divisible by 16, the
given number is divisible by 16 or if there are 4 zeroes in the end, it will be
divisible by 16.
15. Divisibility by 17 : If a given number is of three
or four digits, then subtracting the twice the number formed by thousand’s and hundred’s
digits from the number formed the ten’s
and unit place digits, if we get zero or a number divisible by 17, then the
given number is divisible by 17.
If the number is formed of 2 or 3 digits, then add the two
times of unit’s place digit to three times of number formed by hundred’s and
ten’s place digits. If the sum is divisible by 17, the original number will
also be divisible by 17.
16. Divisibility by 18 : A
number which is divisible by 2 and 9 is divisible by 18.
17. Divisibility by 19 : Add twice the digits at unit’s
place in the remaining number successively, if the sum is divisible by 19, the
given number is divisible by 19.
Example: 323 ( 32+2x3 =32+6 =38 which is divisible by 18, so the number is also divisible by 18.
3249 ( 324+2x9 = 324 + 18 = 342 repeating the process
34 + 2x2 = 34 + 4 = 38 which is divisible by 18 so the given number is also divisible by 18
18. Divisibility by 20 : The number which is divisible by 4 and 5 is also divisible by 25.
19. Divisibility by 25 : In any given number if the number formed by the last two digits is
divisible by 25, the given number is divisible by 25.
Example : 26525, 34375 , 45650, ( These numbers are divisible by 25 because last two digits are also divisible by 25)
6500, 8100. 76500 are also divisible by 25 because the last two digits are zero which is divisible by 25
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