## Generalized Divisibility Rules For Divisibility Tests

How do you find whether a number is exactly divisible by another number? By performing division calculation? No, you cannot do that for every number. That’s why we need to follow divisibility rules. These are useful to solve problems quickly in Aptitude Tests. These rules are mainly useful to check if a number is Prime. Suppose you want to find 45647845631214566461 is divisible by 3 or not. You cannot perform division calculation because it takes a lot of time. This is where you need divisibility rules. This rules makes your division easy. Here, divisibility rules for different numbers with examples under each divisibility rule is explained and generalized rules for divisibility by any number is also included at the end.

## Contents :

1. What is Divisibility
2. Divisibility Rules For Numbers Up To 20
3. Generalized Divisibility Rules

### What is Divisibility :

“Divisible by” means when you divide one number with another number the result should be whole number with zero remainder.

Ex :
6/3 = 2 ; 6 is divisible by 3, because result 2 is whole number and remainder is 0.
7/3 = 2.33 ; 7 is not divisible by 3, because result 2.33 is not whole number and remainder is 1.

Co Primes : H.C.F. of two numbers in one set should be 1.
Ex : (2,3),(4,5),(6,7),(11,15)

If a number N is divisible by two numbers a and b, where a, b are co primes, then N is divisible by ab.

Ex :
24 is divisible by 2 and 3 where 2,3 are co primes, then 24 is divisible by 6.
36 is divisible by 2, 4 where 2, 4 are not co primes, so, 36 is not divisible by 8.

If a number N is divisible by another number M, then N is also divisible by the factors of M.

Ex:
72 is divisible by 12, 72 is also divisible by 2, 3, 4, 6. because 2, 3, 4, 6 are factors of 12.

If a number N is divisible by prime factors of M, then N is also divisible by M.

Ex:
63 is divisible by 3 and 7, 63 is also divisible by 21, because 3 and 7 are prime factors of 21.

### Divisibility Rules For Numbers Up To 20 :

#### Divisibility by 1 :

Every integer is divisible by 1, so no rules are needed.
Ex: 3 is divisible by 1, 4573842 is divisible by 1.

#### Divisibility by 2 :

The last digit must be even number i.e. 0, 2, 4, 6, 8.
Ex : 3456 is divisible by 2  –> last digit i.e. 6 is an even number.
343423 is not divisible by 2  –>  las
t digit 3 is not an even number.

#### Divisibility by 3:

The sum of digits in given number should be divisible by 3.
Ex : 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 :

The number formed by last two digits in given number must be divisible by 4.
Ex : 23746228 is divisible by 4   –>   28 is divisible by 4.
674235642 is not divisibility by 4    –>   42 is not divisibility by 4.

#### Divisibility by 5 :

Last digit must be 0 or 5.
Ex: 42340 is divisible by 5 –> 0 is last digit.
675564 is not divisibility by 5 –> 4 is last digit.

#### Divisibility by 6:

The number must be divisible by 2 and 3. Because 2 and 3 are prime factors of 6.
Ex : 7563894 is divisible by 6 –> last digit is 4, so divisible by 2, and sum 7+5+6+3+8+9+4 = 42 is divisible by 3.
567423 is not divisible by 6 –> last digit is 3, so not divisible by 2. No need to check for 3.

#### Divisibility by 7:

Twice the last digit and subtract it from remaining number in given number, result must be divisible by 7. (You can again apply this to check for divisibility by 7.)
Ex :
1. 343 is divisible 7 –> 34 – (2*3) = 28, 28 is divisible by 7.
2. 345343 is given number. 3 is last digit. Subtract 2*3 from 34534.
34534 – (2*3) = 34528, we cannot tell that this result is divisible by 7. So, we do it again.
3452 – (2*8) = 3436, we will do it again.
343 – (2*6) = 331, we will do it again.
33 – (2*1) = 31, 31 is not divisible by 7.
So, 345343 is not divisible by 7.

Don’t forget to check the generalized rule at the end.

#### Divisible by 8:

The number formed by last three digits in given number must be divisible by 8.
Ex : 234568 is divisible by 8 –> 568 is divisible by 8.
4568742 is not divisible by 8 –> 742 is not divisible by 8.

#### Divisible by 9:

Same as 3. Sum of digits in given number must be divisible by 9.
Ex : 456786 is divisible by 9 –> 4+5+6+7+8+6 = 36 is divisible by 9.
87956 is not divisible by 9 –> 8+7+9+5+6 = 35 is not divisible by 9.

#### Divisible by 10 :

Last digit must be 0.
Ex : 456780 is divisible by 10 –> last digit is 0.
78521 is not divisible by 10 –> last digit is 1.

#### Divisible by 11 :

Form the alternating sum of digits. The result must be divisible by 11.
Ex : 416042 is divisible by 11 –> 4-1+6-0+4-2 = 11, 11 is divisible by 11.
8219543574 is divisible by 11 –> 8-2+1-9+5-4+3-5+7-4 = 0 is divisible by 11.

#### Divisibility by 12 :

The number must be divisible by 3 and 4. Because $$3~\times ~{{2}^{2}}$$ are prime factors of 12.
Ex : 462157692 is divisible by 12 –> last 2 digits 92, so divisible by 4, and sum 4+6+2+1+5+7+6+9+2 = 42 is divisible by 3.
625859 is not divisible by 6 –> last 2 digits 59, is not divisible by 4. No need to check for 3.

#### Divisibility by 13 :

Multiply last digit with 4 and add it to remaining number in given number, result must be divisible by 13. (You can again apply this to check for divisibility by 13.)
Ex :
1. 4568 is not divisible by 13 –> 456 + (4*8) = 488 –> 48 + (4*8) = 80, 80 is not divisible by 13.
2. 593773622 is given number. 2 is last digit. add 4*2 to 59377362 -> 59377370
5937737+ (4*0) = 5937737, we cannot tell that this result is divisible by 13. So, we do it again.
593773 + (4*7) = 593801, we will do it again.
59380 + (4*1) = 59384, we will do it again.
5938 + (4*4) = 5954, we will do it again.
595 + (4*4) = 611, we will do it again.
61 + (4 *1) = 65 , 65 is divisible by 13.
So, 593773622 is divisible by 13.

Don’t forget to check the generalized rule at the end.

#### 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.

#### Divisible by 16:

The number formed by last four digits in given number must be divisible by 16.
Ex : 7852176 is divisible by 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 in divisibility by 7 and divisibility by 13.

#### Divisibility by 18 :

The number should be divisible by 2 and 9. Because $$2~\times ~{{3}^{2}}$$ are prime factors of 18.

#### Divisibility by 19 :

Multiply last digit with 2 and add it to remaining number in given number, result must be divisible by 19. (You can again apply this to check for divisibility by 19.)
Follow the similar examples given in divisibility by 7 and divisibility by 13.

#### Divisibility by 20 :

The number formed by last two digits in given number must be divisible by 20.
Ex : 2374680 is divisible by 20          –>      80 is divisible by 20.
456215789654824 is not divisibility by 20    –>     24 is not divisibility by 20.

### Generalized Divisibility rules :

To check divisibility by a number You should check divisibility by highest power of each of its prime factors.
Remember that the divisibility by any two of the factors of a number is not sufficient to judge its divisibility.

Ex :
1. Suppose you are checking for divisibility by 12. 2,3,4,6 are factors of 12. You cannot say that the number is divisible by 12 by checking divisibility by only 2 and 4 or 4 and 6. You should check divisibility by 3 and 4 to tell the divisibility by 12. Because 3 and 4 are prime factors of 12.
2. Now, we are checking divisibility by 18. Prime factors of 18 are $$2~\times ~{{3}^{2}}$$.  So, you should check divisibility by 2 and 9 . don’t check for just 2 and 3 and also don’t check for 3 and 6.

Basically prime numbers have factors 1 and the number itself. So, we can’t this rule apply for prime numbers. This is only for composite numbers.

#### Divisibility by numbers which ends in 1,3,7,9 :

So, this rule counts for prime numbers which have been missed in previous rule.

To test divisibility by a number N which ends in 1,3,7,9 this method can be used.
Multiply N with any number to get 9 in the end.  Add 1 to the result and divide it by 10.
Store the above result as R.
We are checking whether a number X is divisible by N or not.
Split X as X = 10 y + z ;
X is divisible by N, only if Rz + y is divisible by N.

Ex – 1 :
Find whether 645 is divisible by 23 or not.
N =23 ; 23 * 3 = 69 ; so now N has 9 in the end.
R = (69 + 1) / 10 = 7 ;
X = 645 ; split X as X = 10 y + z ;
645 = (10 * 64) + 5 ;
Y = 64 ; z = 5 ;
Rz + y = (7 * 5) + 64 = 35 + 64 = 99 ;
99 is not divisible by 23. So, 645 is also not divisible by 23.

Ex – 2 :
Let us find 585 is divisible by 39 or not.
N =39 ;  so now N has 9 in the end.
R = (39 + 1) / 10 = 4 ;
X = 585 ; split X as X = 10 y + z ;
645 = (10 * 58) + 5 ;
Y = 58 ; z = 5 ;
Rz + y = (4 * 5) + 58 = 20 + 58 = 78 ;
78 is divisible by 39. So, 585 is also divisible by 39.

## How To Find Prime Numbers and How To Check if a Number is Prime

How to find if a number is prime? We get questions on this topic in some competitive exams. Deciding whether a number is prime number or not is not a simple task if it is bigger number. So, some techniques are needed to tell if the number is prime. This is what you are looking for, then you are in right place. Because I have covered everything you need to know about Prime numbers.

### Prime Number Definition :

A Number which is divisible by 1 and the number itself is called as PRIME NUMBER.
Prime number must be a positive integer and it should be greater than 1.

Ex :

17 is divisible by 1 and 17 only, So, it is prime number.

15 is divisible by 1,3,5,15. So, it is not prime number.

### Prime Factorization – Finding Prime Factors :

Factors of any number should be prime numbers in prime factorization.
Factors are numbers which are multiplied to get original number.
2 * 3  =  6   => 2, 3 are factors of 6. They are also prime factors of 6.
54  = 9 * 6  => 9,6 are factors of 54, but they are not prime factors.

#### Factor Tree :

Factor Tree can be used to get prime factors of a number.

2 * 3 * 3 * 3 = 54 => this is how we do prime factorization.
$$2~\times ~{{3}^{3}}$$ are prime factors of 54.

### Finding all prime numbers between 1 and 100 in a simple way :

25 prime numbers are there in between 1 and 100.
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97…………

1. First arrange numbers in a table like shown in figure.
2. Enter 6 numbers in each row until the last number (in this it is 100) reaches.
4. Round off number 2 and strike off entire column until the end.
5. Similarly strike off 4th column and 6th column as they are divisible by 2.
6. Now round off next number 3 and strike off entire column until end.
7. 4 is gone.
8. Now round off next number 5 and strike off numbers in inclined fashion as shown in figure. When a striking off ends in some row, start again striking off with number in another end  which is divisible by 5 in a row parallel to previous strike off line as shown in figure.
9. 6 is gone.
10. Now round off number 7 and strike off numbers as we did in case of number 5.
11. 8,9,10 are gone.
12. Stop at this point. Wonder why do we need to stop??
13. It’s a trick that we have to do this procedure until we reach square root of bigger number.
14. This technique was developed in 3rd century B.C.

### How To check if a Number is Prime :

Divide the given number with every number below it except 1 and the number itself.
Check whether the given number is divisible by any of the numbers below it.
If it is not divisible by any number then it is prime number.

"Divisible by" means when you divide the whole number with another whole number result should be whole number with remainder zero.

Actually you don’t need to divide with all numbers.

First check whether it is divisible by 2 or not.
Next check with 3, 5, 7, 11,13…..
Check divisibility only with prime numbers.

Here also you don’t need to divide with all prime numbers below it.

Let p be the given number.
Find a number n greater than $$~\sqrt{p}$$
Find prime numbers below n and divide p with only those prime numbers below n.

Actually, we used the same procedure in the above figure of finding all prime numbers upto 100.

Ex : 1 - Number is 149

Let us find whether 149 is prime or not.
$\sqrt{149}= 12.20 < 13$
prime numbers below 13 are 2,3,5,7,11….
149 is not exactly divisible by 2, because it dont has even number at end.
149 is not exactly divisible by 3, because the sum of numbers 1+4+9= 14 is not divisible by 3.
149 is not exactly divisible by 5, because it dont has 0 or 5 at end.
149 is not exactly divisible by 7,11.
So, 149 is prime number…

Ex: 2 - number is 631

Let us find whether 631 is prime or not.
$\sqrt{631}= 25.11 < 26$
prime numbers below 26 are 2,3,5,7,11,13,17,19,23….
631 is not exactly divisible by 2,3,5,7,11,13,17,19,23…
So, 631 is prime number…

For easy divisibility checking, you should know the divisibility rules.

### Some Facts on Prime Numbers:

Co Primes:

H.C.F. of two numbers in one set should be 1.
Ex: (2,3),(4,5),(6,7),(11,15)

If a number N is divisible by two numbers a and b, where a, b are co primes, then N is divisible by ab.

Twin Primes:

Difference between prime numbers in one set should be 2.
Ex: (3,5),(5,7),(11,13),(17,19),(31,33)

If a Number M is a prime number and N is a next prime number, the average difference between M and N is “ ln(M) ” [natural logarithm of M]
• 2 is the only one prime even number.
• Numbers greater than 1 and which are not prime numbers are composite numbers.
• Some probabilistic methods are available for checking big prime numbers.

### Prime Numbers upto 1000 :

Here is the list of prime numbers upto 1000 for your reference.

Generalized divisibility rules

## Marks and Time puzzle 59

A teacher gave 13 marks to one student and 12 marks to another student in one exam.
Can you tell the TIME by using the above sentence ????

1.45
The teacher gave a total of 25 marks to two students. 25 is a quarter.
So, teacher gave “Quarter to Two”.

Time format to “Quarter to Two” is 1.45.

## Birthday Brainteaser 58

The day before yesterday I was 25.
The next year I will be 28.
This is true only one day in a year.
What day is my Birthday ?

My birthday is on December 31. I am telling this on January 1.
Day before yesterday (dec 30)    = I am 25
Present day (January 1)               = I am 26
this year december 31                 = I will be 27.

Next year december 31               = I will be 28.

## Simple Maths Question 57

What Mathematical symbol can be placed in between 3 and 7, to get a number which lies in between 3 and 7???

Exact Answer      = 3.7 (Decimal symbol)
Related Answer  = 3cot7 =3 * 1.14 = 3.42

## Litres Measuring Puzzle 56

You have 3 litre bottle and 5 litre bottle. How can you measure 4 litres of water by using 3Lt and 5Lt bottles???

Solution 1 :

1. First fill 3Lt bottle completely and pour 3 litres into 5Lt bottle.
2. Again fill 3Lt bottle completely. now pour 2 litres into 5Lt bottle until it becomes full.
3. Now empty 5Lt bottle.
4. Pour remaining 1 litre in 3Lt bottle into 5Lt bottle.
5. Now again fill 3Lt bottle completely and pour 3 litres into 5Lt bottle.
6. Now you have 4 litres in 5Lt bottle. That's it.

Solution 2 :

1. First fill 5Lt bottle completely and pour 3 litres into 3Lt bottle.
2. Empty 3Lt bottle.
3. Pour remaining 2 litres in  5Lt bottle into 3Lt bottle.
4. Again fill 5Lt bottle completely and pour 1 litre into 3 Lt bottle until it becomes full.
5. Now you have 4 litres in 5Lt bottle. That's it.

## Missing Rupee Brain Teaser 55

3 Friends went to a shop and purchased 3 toys. Each person paid Rs.10 which is the cost of one toy. So, they paid Rs.30 i.e. total amount. Shop owner gave a discount of Rs.5 on the total purchase of 3 toys for Rs.30. Then, among Rs.5, Each person has taken Rs.1 and remaining Rs.2 given to the beggar beside the shop.
Now, the effective amount paid by each person is Rs.9 and the amount given to beggar is Rs.2. So, total effective amount paid is 9*3 = 27 and the amount given to beggar is Rs.2, thus the total is Rs.29. Where has the other Rs.1 gone from the original Rs.30 ?????

The logic is payments should be equal to receipts. we cannot add amount paid by persons and amount given to beggar and compare it to Rs.30.
The total amount paid is Rs.27. So, from Rs.27, shop owner received Rs.25 and beggar received Rs. 2. Thus, payments are equal to receipts.

## Magic Room Riddle 54

You are in one magic room. That room is having two doors. one door leads to death and another door is safe. If you ask a question, one door always tells truth and another door always lies and you don't know which one tells truth and which one lies. You can ask only one question to only one door and you must get through the safe door. So, how will you manage to get through the safe door ?? Which question would you ask and To which door you would ask ????

Answer : Just go to any one door and ask, "what the other door would tell if you asked which door is safe?" and go through another door. That's it. So, the question is important and the door to which you are asking doesn't matters.
Still confused.. Let me know it in comments...

## Number of Days Puzzle 53

A man is climbing up a mountain which is inclined. He has to travel 100 km to reach top of mountain. Every day He climbs up 2 km forward in day time. Exhausted, he then takes rest there at night time. At night, while he is asleep, he slips down 1 km backward because mountain is inclined.
Then how many days does it take him to reach mountain top ????

Solution : Each day, Total progress = 2 km - 1 km = 1 km
So, 98 days = 98 Kms.
On 99th day he can reach mountain top by travelling 2 km in day time.

## Decision Making PUZZLE 52

For Rs.1 You get 40 Bananas.
For Rs.3 you get 1 Mango.
For RS.5 you get 1 Apple.

Now you want to get 100 fruites for Rs.100.

So, How many Bananas, Mangoes and Apples will you buy ??

Rs.95 - 19 Apples
Rs.3  - 1 Mango
Rs.2  - 80 Banans

Total 100 Fruites for RS.100.

## Basic Number Theory for Problems on numbers in Aptitude

Hi everyone, you may not get direct questions on this in any aptitude type test. But, by knowing this number system basics you can be able to solve some silly questions on numbers in some simple tests. So, Get your basics brushed up with this.

0,1,2,3,4,5,6,7,8,9      -    Digits
group of digits             -    Numbers

Natural numbers                    :    1,2,3,4,5,6,7,8,9,10,11………………….
Whole numbers                     :    0,1,2,3,4,5,6,7,8,9,10,11,12…………….
Integers                                 :    …………-4,-3,-2,-1,0,1,2,3,4………………….
Positive integers                     :   1,2,3,4,5,6,7………….
Negative integers                   :   ………….-7,-6,-5,-4,-3,-2,-1
Non positive integers              :   0,-1,-2,-3,-4…….
Non negative integers             :   0,1,2,3,4,5,6…………………

0 is neither positive nor negative.

Even numbers                :         A number divisible by 2
2,4,6,8,10,12,14,16,18,20………………
Odd numbers                 :         A number not divisible by 2.
1,3,5,7,9,11,1,3,15,17,19,21……………………
Prime numbers               :         A number having exactly 2 factors, namely 1 and the number itself.
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97………
2 is the only one prime even number.
25 prime numbers are there in between 1 and 100.
Composite numbers        :          Numbers greater than 1 which are not prime numbers.
4,6,8,9,10,12,14,15…………….
0,1 are neither prime nor composite.
Co primes                       :           H.C.F. of two numbers should be 1.
(2,3),(4,5),(6,7),(11,15)
Twin primes                     :           Difference between prime numbers in one set should be 2.
(3,5),(5,7),(11,13),(17,19),(31,33)

If a number N is divisible by two numbers a and b, where a, b are co primes, then N is divisible by ab.
Ex :  72 is divisible by 2 and 3, where (2,3) are co primes. So, 72 is divisible by 6 i.e. (2×3)

Face value         :    The actual value of a digit in the given number
Place value        :    This value varies depends upon place of digit in a given number.
Ex :  In 895473,
Face value of 3 is 3 , face value of 9 is 9.
Place value of 3 = 3*1 = 3
Place value of 7 = 7*10 = 70
Place value of 4 = 4*100 = 400
Place value of 9 = 9 * 10000 = 90000

Division rule :
Dividend = (Divisor × Quotient) + Remainder

Ex :  14 = (3 × 4) + 2
14  =   Dividend
3   =   Divisor
4   =   Quotient
2   =   Reminder

check how to solve section in top menu bar

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