### Generalized Divisibility Rules

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**

### 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 we have covered everything you need to know about Prime
numbers.

### Contents

####
1. Prime Number Definition
2. Prime Factorization – Finding Prime Factors
3. Finding all prime numbers between 1 and
100 in a simple way
4. How to check if a number is Prime
5. Some facts about prime numbers
6. prime numbers upto 1000

### 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.

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

This technique can be used to find all prime numbers between any two numbers25 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…………

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

### How To check if a Number is Prime

Divide the given number with every number below it.

Check whether the given number is divisible by any of the numbers below it except 1.

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…..

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.

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….

\[\sqrt{149}= 12.20 < 13\]

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…

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….

\[\sqrt{631}= 25.11 < 26\]

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.

Read about generalized divisibility rules here

###
Some Facts on Prime Numbers

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

Thats all about Prime Numbers.

Read again about Basic Number theory

Generalized divisibility rules

check about blood relations tricks

So, 631 is prime number…

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

Read about generalized divisibility rules here

###
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.

Thats all about Prime Numbers.

Read again about Basic Number theory

Generalized divisibility rules

check about blood relations tricks

### Marks and Time puzzle 59

### 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

### 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.

check how to solve blood relations problems easily.

### 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.

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.

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