Divisibility by 7
-> To find out if a number is divisible by seven, take the last digit, double it, and
subtract it from the rest of the number, if the resulting number is divisible by 7
then original number is divisible by 7. If you don't know the new number's
divisibility, you can apply the rule again.
Example:
Which of the following numbers is/are fully divisible by 7: 203, 3192, 3197,
38241?
Answer : 203, 3192, 38241
Reason:
For 203, you would double the last digit to get six, and subtract that from 20 to get
14. Hence 203 is divisible by 7.
Applying the same rule to check if 3192 is divisible by 7:
3192 => 319 - 2*2 = 315 => 31 - 2*5 = 21, 21 is divisible by 7 hence 3192 is
divisible by 7
Applying the same rule to check if 3197 is divisible by 7:
3197 => 319-2*7 = 305 => 30 -2*5 = 20, 20 is not divisible by 7 hence 3197 is
not divisible by 7
Applying the same rule to check if 38241 is divisible by 7:
38241 => 3824 -2*1 = 3822 => 382 - 2*2 = 378 => 37 -2*8 = 21, 21
is divisible by 7 hence 38241 is divisible by 7
Divisibility by 8
-> A number is divisible by 8 if and only if the last 3 digits of a number divisible by
8.
Example:
Which of the following numbers is/are fully divisible by 8:
97533224, 97533328, 97533222, 97533228?
Answer: 97533224 and 97533328
Reason: 224 and 328 are divisible by 8 whereas 222 and 228 are not divisible by
8.
Divisibility by 9
-> A number is divisible by 9 if and only if the sum of the digits is divisible by 9.
Example:
Which of the following numbers is/are fully divisible by 9:
111111111, 111222, 2222244, 66669 699999?
Answer: 111111111, 111222 and 2222244
Reason: Sum of digits for 111111111, 111222 and 2222244 are 9, 9 and
18 respectively all of which are divisible by 9. Whereas sum of digits for 66669
& 699999 are 33 & 51 respectively both of which are not divisible by 9.
Divisibility by 10
-> A number is divisible by 10 if and only if the number ends in n zeros.
Example: If a number ends in 0 then it would be divisible by 5. Such as 111110,
876543200
Divisibility by 11
-> A number is divisible by 11 if the difference of the sum of digits at odd places
and the sum of its digits at even places, is either 0 or divisible by 11, then clearly
the number is divisible by 11.
Example:
Which of the following numbers is/are fully divisible by 11: 6050, 3035362, 90002,
900002?
Answer: 6050, 3035362, 90002
Reason: Sum of digits at even place in 6050 = 11 and sum of digits at odd place
is 0. Difference between odd and even = 11. Hence 6050 is divisible by 11.
Similarly sum of digits at even place in 3035362 = 11 and sum of digits at odd
place is 11. Difference between odd and even = 0. Hence 3035362 is divisible by
11.
Similarly Difference between odd and even digits for 90002 is 11 whereas for
900002 difference between odd and even digits is 7. Hence 90002 is divisible by 11
but 900002 is not.
Divisibility by 12
-> A number is divisible by 12 if the number is divisible by both 3 and 4
Example:
Which of the following two numbers is/are divisible by 12: 22584 and 22756?
Answer: 22584
Reason: 22584 is divisible by 3 because sum of all digits =21 which is divisible by 3
and 22584 is also divisible by 4 because last 2 digits of 22584 i.e. 84 is divisible by
4. Hence 22584 is divisible by 12
22756 is not divisible 3 as sum of digits are 22 which is not divisible by 3.
Divisibility by 13
-> To find out if a number is divisible by 13 take the last digit of number and
multiple it by 4 and add it to number formed with remaining digits check if the
resultant number is divisible by 13. In case resultant number is big repeat process
of multiplying last digit by 4 and adding it to number formed by remaining digits.
Example:
Which of the following numbers is/are divisible by 13: 11013, 110006, 110026 ?
Answer: 110006
Reason: Applying the rule above of multiplying last digits by 4 and adding it to
number formed by remaining digit we get : 1100+4*4= 11024
Continuing the same way with resultant number we get : 1102+4*4= 1118
=> 1118 = 111+8*4 = 143
=> 14+3*4 = 26. 26 is divisible by 13. Hence 110006 is divisible by 13
11013 is not divisible by 13 as per results below:
11013= 1101+3*4 = 1113 => 1113 = 111+3*4 = 123 => 123 = 12+3*4 =24
24 is not divisible 13 hence 11013 is not divisible by 13.
-> If n is even , n(n+1)(n+2) is divisible by 24