TESTS OF DIVISIBILITY

(J). Divisibility by 6 :

A number is divisible by 6 only when it is divisible by 2 and 3 both. So first of all we see that the number is even or not then we check for the divisibility by 3 .

Example ; (1) check whether 216 is divisible by 6 or not .
Solution : step (a). First of all we check the given number is divisible by 2 or not
Since unit digit is divisible by 2 then the given number is divisible by 2 .

Step (2). We see that the given number is divisible by 2 so now we check whether the given number is divisible by 3 or not .
Sum of digits of the given number is (6+1+2) 9 , which is divisible by 3 so the given number is divisible by 3 also .

Step(3). Since the given number is divisible by 2 and 3 both so the given number is divisible by 6 also .

(K). Divisibility by 12 :

 A number is divisible by 12 only when it is divisible by 4 and 3 both at the same time . so first of all check the divisibility by 4 then by 4 .

NOTE : Thus we can conclude that any number which is divisible by a composite number( The natural numbers which are not prime , are called composite numbers) ,as mentioned above , must be divisible by all its factors whose L.C.M is the given number divisior . 

                     TESTS OF DIVISIBILITY

(f).  Divisibility By 10 :

A number is divisible by 10 only when its unit digit is 0(zero).
Example : (1) 68720 is divisible by 10 because its unit digit is 0 .

(2) 29850 is divisible by 10 because its unit digit is 0.

(3) 687405 is not divisible by 10 ,since its unit digit is 5 not 0 .

(G).  Divisibility by 5 :

A number is divisible by 5 only when its unit digit is 0 or 5 .

Example ; (1) 6785 is divisible by 5 because its unit digit is 5 .
(2) 96750 is divisible by 5 because its unit digit is 0 .

(H). Divisibility by 7,11,13 :
A number can be divisible by 7,11 or 13  if and only if the difference of the number formed by the last three digits and the number formed by the rest digits is divisible by 7,11 or 13 respectively .

Example ; check whether 139132 is divisible by 7 or not .

Solution; step (H.1)
 number formed by the last three digits is 132 .
Step (H.2) : number formed by rest of digits is 139
Step (H.3) ; we take difference between these two numbers
139 - 132 = 7
Since the difference is divisible by 7 . Hence the given number is also divisible by 7.

(2) check whether 12478375 is divisible by 13 or not.

Solution;  step1 . 12478 - 375 = 12103
                  Step 2. 12 - 103 = - 91
Since the difference is divisible by 13 .Hence the given number is divisible by 13 .

(3) check whether 29435417 is divisible by 11or not.

Solution ; step(1) 29435 - 417 = 29018
                   Step(2) 29 - 018 = 11
Since the difference is divisible by 11 so the number is also divisible by 11 .

(I). Divisibility by 11 :
A number is divisible by 11 if the difference between the sum of the digit at odd places and sum of digits at even places is equal to zero or a number divisible by 11.

Example ; check 29435417 is divisible by 11 or not .
Solution; step(1). Sum  of its digits at odd places is
7+4+3+9 = 23
                  Step(2). Sum of its digits at even places is
1+5+4+2 = 12
                 Step (3). Difference between these two numbers is
23 - 12 = 11
Since difference is divisible by 11 .Hence the given number is also divisible by 11 .

(2). 57945822
(Sum of digit at odd places - sum of digits at even places)
(2+8+4+7) - ( 2+5+9+5) = 21 - 21 = 0
Since the difference is zero so the number is divisible by 11 .

                          TESTS OF DIVISIBILITY
(a) Divisibility By 2 :
A number is divisible by 2 if its unit digit is divisible by 2 .
Example : (1). 7684
Solution:
The unit is 4 , which is divisible by 2 (4÷2=2)  so 7684 is divisible by 2.
(2). 2785
Solution : The unit digit is 5 , which is not divisible by 2 . so 2785 is never divisible by 2.

(b). Divisibility by 3 :
A number is divisible by 3 only when the sum of its digit is divisible by 3.
Example : (1). 695421
Solution ; in the number 695421 , the sum of the digit is 27 , which is divisible by 3.
So the number is divisible by 3 .
(2). 948655
Solution : in the number 948655 , the sum of the digits is 37 , which is not divisible by 3 .
So the number 948655 is not divisible by 3 .

(c). Divisibility by 9 :
A number is divisible by 9 only when the sum of its digits is divisible by 9 .
Example : (1). 246591
In the number 246591 the sum of its digits
= 27 , which is divisible by 9.
So the number is divisible by 9 .

(d) . Divisibility by 4 : A number is divisible by  4 if the number  formed by its last two digits is divisible by 4 .
Example : (1) 6879376
Solution : since 76 {the last two digit} is divisible by 4 . so the number 6879376 is divisible by 4 .

(e). Divisibility by 8 : A number is divisible by 8 if the number formed by hundred's ten's and unit's digit of the given number is divisible by 8  .
Example : (1) 16789352 , the number formed by last three digits , namely 352 is divisible by 8. So 16789352 is divisible by 8 .

Numbers:in hindu - Arabic system , we have ten digits , namely 0,1,2,3,4,5,6,7,8,9
Called zero,one,two,three,four,five,six,seven,eight,and nine respectively.
Numeral: A number is denoted by a group of digit , called numeral .
For denoting a numeral ,we use the place - value chart .
2. Face value : The face value of a digit in a numeral is its own value , at whatever place it may be . example
In the numeral 58932
The face value of  9 is 9
The face value of 2 is 2.

3. Types of Numbers :
a. Natural Numbers
b.Whole Numbers
c.Integers

4. Even And Odd Numbers :

5; prime Numbers : A counting number is called a prime number if it has exactly two factors namely itself and 1 that is prime number are divided by 1 and itself ,  not any other number.
All prime number less than 100 are:
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.

Test For a Number to be primes :
Let p be a given number  and let n be the smallest counting number such that (n×n) greater than equal to p {(n×n)>=p} .
Now test whether p is divisible by any of the prime number less than or equal to n.
Example : (1) test which of the following are prime numbers?
(A) 61
Solution : Step (A.1)
choose a number whose square is greater than the given number (like in this question 61)
The closest number n= 8
Step(A.2):  now square of 8 is
n×n = 8×8= 64
which is greater than 61

Step (A.3) : prime number less than 8 is 2,3,5,7
Step (A.4) : now 61 is divided by  these numbers  we see that none of them divides 61
Hence 61 is a prime number

(B) : 173
n = 14
14×14 = 196 which is greater than 173
Prime numbers less than 14 is 2,3,5,7,11,13
Clearly , non of them divides 173
Hence it is a prime number

(C) : 321
n = 18
18×18 = 324, which is greater than 321
Prime numbers less than 18 are 2,3,5,7,11,13,17,
We see that 3 divides 321 hence 321 is not a prime number .
Other questions are
(D): 437, 137 , 991 etc .