DIVISIBILITY RULES
Divisibility Rules for 11.
If the difference of the sum of alternate digits of a number is divisible by 11, then that number is divisible by 11 completely.
i.e., Sum of digits in odd places – Sum of digits in even places = 0 or a multiple of 11
Here is the following procedure
Group the alternative digits i.e. digits which are in odd places together and digits in even places together. Here 24 and 13 are two groups.
Example: Is 2143 divisible by 11?
Sol : Sum of digits in even places = 4 + 2 = 6
Sum of digits in odd places = 3 + 1 = 4
Here 2 is the difference which is not divisible by 11.
Therefore, 2143 is not divisible by 11.
I. If the number of digits of a number is even, then add the first digit and subtract the last digit from the rest of the number.
Example: 3784
Number of digits =4
Now, 78 + 3 – 4 = 77 = 7 × 11
Thus, 3784 is divisible by 11
II. If the number of digits of a number is odd, then subtract the first and the last digits from the rest of the number.
Example: 82907
Number of digits =5
Now, 290 – 8 – 7 = 275 = 25 x 11
Thus, 82907 is divisible by 11
Divisibility Rules for 12:
If the number is divisible by both 3 and 4, then the number is divisible by 12 exactly.
Example: 5864
Sum of the digits = 5 + 8 + 6 + 4 = 23 (not a multiple of 3)
Last two digits = 64 (divisible by 4)
The given number 5864 is divisible by 4 but not by 3; hence, it is not divisible by 12
Divisibility Rules for 13
For any given number, to check if it is divisible by 13, we have to add four times of the last digit of the number to the remaining number and repeat the process until you get a two-digit number. Now check if that two-digit number is divisible by 13 or not. If it is divisible, then the given number is divisible by 13
For example: 2795
279 + (5 x 4) = 279 + (20) = 299
29 + (9 x 4) = 29 + 36 = 65
Number 65 is divisible by 13, 13 x 5 = 65
Note
1. If a number is divisible by 15, it is divisible by both 5 and 3.
2. If a number is divisible by 18, it is divisible by both 9 and 2
3. If a number is divisible by 25, the number formed by last two digits is divisible by 25
4. If a number is divisible by 45, it is divisible by both 9 and 5
5. If a number is divisible by 72, it is divisible by both 8 and 9.
Division Algorithm
The division algorithm says when a number 'a' is divided by a number 'b' gives the quotient to be 'q' and the remainder to be 'r' then a = bq + r where 0 r < b. This is also known as "Euclid's division lemma". The division algorithm can be represented in simple words as follows
Dividend = Divisor × Quotient + Remainder
Let us just verify the division algorithm for some numbers. We know that when 59 is divided by 7, the quotient is 8 and the remainder is 3. Here
dividend = 59
divisor = 7
quotient = 8
remainder = 3
Verification of division algorithm:
Dividend = Divisor × Quotient + Remainder
59 = 7 × 8 + 3
59 = 56 + 3
59 = 59
Hence, the division algorithm is verified.