Hello Friends!
Today we will discuss a very important topic for the Quantitative Aptitude part that is how to deal with Divisibility Rules. The problem on Divisibility is one of the common topics in SSC, IBPS, CAT and other Exams. So, these short notes will be helpful for your upcoming Exams.
Remember
Today we will discuss a very important topic for the Quantitative Aptitude part that is how to deal with Divisibility Rules. The problem on Divisibility is one of the common topics in SSC, IBPS, CAT and other Exams. So, these short notes will be helpful for your upcoming Exams.
Remember
- All whole numbers are divisible by 1.
- A number is divisible by 2 if it's even.
- A non-zero number is divisible by 5 if its ends in 0 or 5.
- In order to check the divisibility of a number by a composite number, divide the composite divisor into prime factors, which are co-prime and then check for its divisibility with each.
For Example: To check the divisibility of a number with 12, break down 12 into 3 and 4.
Divisibility Rules
Divisible by means when you divide one number by another number the result should be the whole number with zero remainders.
Example:-
6/3 = 2; 6 is divisible by 3 because result 2 is the whole number and the remainder is 0.
7/3 = 2.33; 7 is not divisible by 3 because result 2.33 is not the whole number and the remainder is 1.
Divisibility by 2: Any number, the last digit of which is rather even or zero, is divisible by 2.
Example:-
1875494: The last digit is 4 and it is an even number so this fat number 1875494 is divisible by 2.
Divisibility by 3: If the sum of the digits of a number is divisible by 3, the number is also divisible by 3.
Example:-
- 3789 is divisible by 3
Sum 3+7+8+9 = 27 is divisible by 3.
- 43266737 is not divisible by 3
Sum 4+3+2+6+6+7+3+7 = 38 is not divisible by 3.
Divisibility by 4: A number is divisible by 4 if the number's last two digits are divisible by 4.
Example:-
- 112 since the last two digits, 12, are divisible by 4, the number 112 satisfies this rule and is also divisible by 4.
- 10,948 the last two digits, 48, are divisible by 4. Therefore, the whole number is also.
Divisibility by 5: If a number ends in 5 or 0, the number is divisible by 5.
Example:-
- 1345: As its last digit is 5, it is divisible by 5.
- 1340: as its last digit is 0, it is divisible by 5.
Divisibility by 6: A number is divisible by 6 if it is even and if the sum of its digits is divisible by 3.
The rule for checking whether a number is divisible by 6 is quite tricky.
Since 6=2*3
We must apply the rules of 2 and 3 on a number to check if it is divisible by 6. So rules are
- The last digit should be even number (Divisibility trick for 2)
- Add up all the digits and it should be multiple of 3 (Divisible by 3)
Example:
2742204: The last digit in 2742204 is 4 and it is the even number.
Add up all the digits so 2+7+4+2+2+0+4 = 24. 24 is divisible by 3.
This number 2742204 satisfy both the rules (2&3) so it is divisible by 6.
Divisibility by 7: Twice the last digit and subtract it from remaining number in given number, the result must be divisible by 7.
Example:
- 343 is divisible 7
34 - (2*3) = 28, 28 is divisible by 7.
Divisibility by 8: If the last three digits of a number are divisible by 8, the number is also divisible by 8, the number is also divisible by 8. Also, if the last three digits of a number are zeros, the number is divisible by 8.
Example:
- 1256: As 256 is divisible by 8, the number is also divisible by 8.
- 135923120: as 120 is divisible by 8, the number is also divisible by 8.
Divisibility by 9: A number is divisible by 9 if the sum of the digits is an evenly divisible 9
Example:
- 39681: 3+9+6+8+1 = 27 is divisible by 9, hence the number is also divisible by 9.
Example: What least whole number should be added to 532869 to make it divisible by 9?
Solution: If a number is divisible by 9, the sum of its digits must be a multiple of 9.
Here, 5+3+2+8+6+9 = 33, the next multiple of 9 is 36.
3 must be added to 532869 to make it divisible by 9.
Divisibility by 10: The last digit must be 0
Example:
- 456780 is divisible by 10. The last digit is 0.
- 78521 is not divisible by 10. The last digit is 1.
Divisibility by 11: If the sums of digits at odd even places are equal or differ by a number divisible by 11, then the number is also divisible by 11.
Example:
3245682
A1 = 3+4+6+2 = 15 and A2 = 2+5+8 = 15
As A1 = A2, the number is divisible by 11.
Example: What is the least value of x such that 7648x is divisible by
Solution: A number is divisible by 11 When difference between the sum of digits at even places and at odd places is 0 or multiple of 11
The given number is 4876x.
(Sum of digits at Even places) - (Sum of digits at ODD places) = 0
(6+8) - (x + 7 + 6) = 0
14 - (X+13) = 0
Here the value of x must be 1.
Divisibility by 12: 12 = 3*4 so if the number follows the divisibility rules of 3 and 4 is divisible by 12.
- Add up all the digits and it should be multiple of 3 (divisible by 3)
- The last 2 digit (ones and tens should be divisible by 4)
Example:
- 834864: Add up all the digits: 8+3+4+8+6+4 = 33 and it is multiple of (3*11)/3
The last two digits are 64 and it is divisible by (4*16)/3 so this number 834864 is divisible by 12.
Divisibility by 13: Multiply last digit with 4 and add it to remaining number in given number, the result must be divisible by 13.
Example:
- 4568 is not divisible by 13
456 + (4*8) = 488
48 + (4*8) = 80, 80 is not divisible by 13.
Divisibility by 14: The number must be divisible by 2 and 7. Because 2 and 7 are prime factors of 14.
Divisibility by 15: The number should be divisible by 3 and 5. Because 3 and 5 are prime factors of 15.
Divisibility by 16: The number formed by last four digits in given number must be divisible by 16.
Example:
7852176 is divisible be 16
2176 is divisible by 16
Divisibility by 17: Multiply last digit with 5 and subtract it from remaining number in given number, result must be divisible by 17. (You can again apply this to check for divisibility by 17)
Follow the similar examples given divisibility by 7 and divisibility by 13.
Divisibility by 18: Any number which is divisible by 9 and has its last digit even or zero, is divisible by 18.
Example:
926565: digit - sum is a multiple of nine (i.e. divisible by 9) and unit digit (8) is even, hence the number is divisible by 18.
Divisibility by 19:
- Double the last digit.
- Add to the rest of the numbers until you get two or three digit number that should be divisible by 19
Example:
6688 8*2 = 16, Add 16 to the rest of the number so 668 + 16 = 684
Repeat it again: 4*2 = 8 and add to 68+8 = 76 and 76 is divisible by 19 so this number 6688 is divisible by 19 as well.
Divisibility by 20:
- The last digit should be 0 (divisible by 10)
- The second last digit should be even
Example:
4490
- The last digit is 0 but the last 2nd digit 9 is not an even number so 4490 is not divisible by 20.
- The last 2 digit should be divisible by 20.
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