HOW TO KNOW THE DIVISIBILITY TEST OF A NUMBER || DIVISIBILITY RULES FOR NUMBERS

How to know whether a given number, however large is divisible by 2, divisible by 3,4,5,6 and 10 .There are fixed divisibility test and divisibility rules for checking the divisibility of any given numbers. For different numbers there are  different rules to divide with . So in this post  we are going to discuss these divisibility rules with examples  one by one. Although there are divisibility test  for fraction also ,but this post will be restricted to natural numbers.

Divisibility rules for 2

To check whether the given number, however large  is divisible by 2 ,we have to check its right most digit/Unit place digit, if it is even number or zero.Then the given number is definitely divisible by 2

For Example

12 is divisible by 2 as its right most digit is 2 which is Even .

3548 is divisible by 2 as its right most digit is 8 which is Even 

999998 is divisible by 2 as its right most digit is 8 which is Even .

65989564 is divisible by 2 as its right most digit is 4 which is Even. 

22222229 is not divisible by 2 as its right most digit is 9 which is not A Even number.

589423100780      is divisible by 2 as its right most digit is 0 .

357913579571536   is divisible by 2 as its right most digit is 6 which is Even although all the remaining digits are odd.

Divisibility rules for 3

To check whether the given number, however large  is divisible by 3,we have to check the SUM of all its digits, if its sum is divisible by 3 then the  given number is divisible by 3, If sum of  all the digits of given number is again a large number then add the result so obtained and apply the rule again which is said earlier.

For Example

15 is divisible by 3 as sum of all its digits 1 + 5 = 6 is divisible by 3 .

833 is not divisible by 3 as sum of all its digits 8 + 3 + 3 = 14 = 1 + 4 = 5 is not divisible by 3 .

2678 is not divisible by 3 as sum of all its digits 2 + 6 + 7 + 8 = 23 = 3 + 3 = 5 is not divisible by3 .

98784552 is not divisible by 3 as sum of all its digits 9 + 8+ 7+ 8+ 4 + 5 + 5 + 2 = 48 = 4 + 8 = 12 = 1 + 2 = 3 is not divisible by 3.

359875269 is divisible by 3 as sum of all its digits 3 + 5 + 9 +8 +7 + 5 + 2 + 6 + 9 = 54 = 5 + 4 = 9 is divisible by 3.

3597841137 is divisible by 3 as sum of all its digits 3 + 5 + 9 + 7 + 8 + 4 + 1 + 1 + 3 + 7 = 48 = 12   is divisible by 3 .

Divisibility rules for 4

If the last two digits of a number is divisible by 4 ,Then the number is divisible by 4 . The number having two or more zeros at the end is also divisible by 4.

For Example

568928 : Here in this number last two digits are 28 ,which are divisible by 4,Hence the given number is divisible by 4.

134826900 : As ther are two zeros at the end,so the given number is divisible by 4.

13444255452 : As the last two digits number (52) is divisible by 4 ,the given number is divisible by 4.

35888875698549 : As the last two digits number (49) is not divisible by 4 ,the given number is not divisible by 4

97971349999567776 : As the last two digits number (76) is divisible by 4 ,the given number is divisible by 4.

44444444444444444449 : As the last two digits number (49) is not divisible by 4 ,the given number is not divisible by 4.

Divisibility rules for 5

To check whether the given number, however large is divisible by 5 ,we have to check its right most digit/Unit place digit,if it is 5 or zero. Then the given number is definitely divisible by 5.I.e if the given number ends with 0 or 5 then it is divisible by by 5.

For Example

35 is divisible by 5 as its right most digit is 5.

97835 is divisible by 5 as its right most digit is 5.

6854940 is divisible by 5 as its right most digit is 0.

35000000355 is divisible by 5 as its right most digit is 5.

3579515465855 is divisible by 5 as its right most digit is 5.

12345678888880 is divisible by 5 as its right most digit is 0.

568954975311525 is divisible by 5 as its right most digit is 5.

55555555555556 is not divisible by 5 as its right most digit is 6.

66666666666666665 is divisible by 5 as its right most digit is 5.

Divisibility rule for 6

For any number to be divisible by 6 ,it must be divisible by both 2 and 3 ,then the given number is divisible by 6, Therefore
1) The number should ends up with an even digits or zero and
2) The sum of its digit should be divisible by 3.


For Example :

56898 is divisible by 6 as sum of its digits is 5 + 6 + 8 + 9 + 8 = 36 = 3 + 6 = 9 so it is divisible by 3 and last digit is even ,as the given number is divisible by both 2 and 3 ,so it is divisible by 6.

3578952  As the last digit is even so the given number is divisible by 2 and sum of all its digits is 3 + 5 +7 + 8 + 9 + 5 + 2 = 39 = 4 + 2 = 6  which is also divisible by 3 ,which implies the given number is divisible by both 2 and 3. Therefore the given number is divisible by 6 as well.

25689879798 is divisible by 6 as sum of its digits is 2 + 5+ 6 + 8 + 9 + 8 + 7 + 9 + 7 + 9 + 8 =78 = 15 = 1 + 5 = 6 so it is divisible by 3 and last digit is even ,as the given number is divisible by both 2 and 3 ,so it is divisible by 6.

35658999962 is not divisible by 6 as sum of its digits is 3 + 5 + 6 +  5 + 8 + 9 + 9 + 9 + 9 + 6 + 2 = 71 = 7 + 1 = 8 so it is not divisible by 3 and last digit is not even ,so it is not divisible by 6.

35789248956 As the last digit is even so the given number is divisible by 2 and sum of all its digits is 3 + 5 + 7+ 8+ 9+ 2+ 4 + 8+ 9 + 5 + 6 = 66 =  6 + 6 = 12 = 1 + 2 = 3 which is also divisible by 3 ,which implies the given number is divisible by both 2 and 3. Therefore the given number is divisible by 6 as well.

Divisibility rule for 10

It is the most easiest number to identify whether it is divisible by 10 , if the given number however large ends up with 0 then it is divisible by 0.

For Example

4546546546540 ,44545454560, 445474456110 5555598959550 are divisible by 10 as all the given numbers ends with 0.

and 445645489,454545,456445555 ,454545577 and 545454 are not divisible by 10 as all these numbers do not ends up with 0.

Conclusion

Thanks for giving your valuable time to read this post of the divisibility rules and divisibility test rules for  divisibility rule for 2 , divisibility rule for 3,    divisibility rule of 4 etc . If  you liked this post  , Then share it with your friends and family members . You can also read my others articles on Mathematics Learning and understanding Maths in easy ways.

Let us learn Multiplication , division , arithmatic and simplification short cut ,tips and  tricks after buying this Book of Magical Mathematics .

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HOW TO MULTIPLY TWO DIFFERENT NUMBERS IN FASTEST AND QUICKEST WAYS

        
  Whenever we are to multiply two numbers then most of the time we undergo long calculations , and if we have to do easy calculation then we were lucky.

But what to do when we have to multiply two numbers in very short time. Suppose we have to multiply 32 with 11 then it is easy, because just put right most digit as it is and then increase every digit by one to left and at last put the left most digit as it is and put 352 as answer.

Example

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SHORTEST METHOD OF MULTIPLICATION FOR TWO NUMBERS

We are going to learn shortest method of multiplication for two numbers which are very close to 100 ,how to multiply two numbers quickly, which can save our time in competitive exams .

Let us understand this multiplication shortcut with the help of this video.

Let us choose two numbers   98 and  96 nearer to 100 
98*96 = (98-4)(2*4) = (94)(08)=9408 (Only in mind )
Step 1 

Consider both the numbers as Num 1 = 98 and Num2 = 96.

Subtract Num 1 from 100 and write its Result 1 as one place .

Also Subtract Num 2 from 100 and write its as Result 2 second place .

Here Result 1 = 100 - 98 = 2
Result 2 = 100 - 96 = 4
Now multiply the results so obtained and mark it as Stepresult1.Stepresult1 = 4*2 = 8 = 08

Step 2 

Subtract the result of  Result 1 (blue Answer) from Num 2 i. e.(96 ).

Here we have
Stepresult2 = 96-2= 94

Final Answer = (Ist two digits are Stepresult2)(2nd two digits are Stepresult1)

= 9408
The result so obtained i. e. 9408 is the answer of product of two numbers

Let us consider  one more example   to fast way to multiply two digit numbers.

Another Example

92 * 91 = ?

Step 1

Last two digits of the answer = (100-92)(100-91)
= 8* 9 = 72

Step 2

1st two digits of the answer = (Greater number as Red highlighted in last step - Smaller number in last step Blue highlighted = 92 - 9 = 83

Final Answer = (Answer of Step 1)(Answer of Step 2) = 8372

 Example   89 * 92 = (92-11)(11*8)=8188

Let us discuss step by step the process of  multiplying these two numbers quickly

Step 1

Split Both the numbers in such a way that one the term must be 100 and other be 100-1st number. i. e. 89 will be written as 100-11 and 92 will be written as 100-8

Step 2

Now select either of the given number (suppose 92 ) and subtract the number ( other than 100 ) which is in other factor in previous step i. e. 11.

Step 3

The difference of 92-11 will the 1st two digits of our answer and product of other two digits are the 3rd and 4th digits of the answer.

One more Example

88×93 = (Answer of Step 1)(Answer of Step 2)
               =   (93-12)  [(100-88) * (100-93)]
            = (93-12)[12*7]
             = (81)(84)
               =8184
94×84 =   (As 94 is 6 less than 100)(As 84 is 16 less than 100)
             = (subtract 16 from 94)(Multiply 6 with 16)
            = (94-16 gives 1st two digits of product) (96 will be last two      digits of product)
Final answer = 7896
You can also learn it as 
93 * 87 = (Greater number - How much second number closer to 100 ) ( (100-93)(100-87))

93 * 87 = (93-13)(7*13)

93*87 = 8091

Conclusion

This method was how to multiply two numbers easily and to multiply two numbers using a shortcut . Thanks for spending your precious time to read this post ,If you liked this post . Please share it with your friends and also follow me on my blog to encourage me to do better than best  for multiplication,  multiplication tricks and   
vedic maths tricks .See your in next post, till then Bye.....

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HOW TO MULTIPLY A NUMBER BY 11 USING VEDIC MATHS TRICKS

Let us discuss the how to multiply two numbers quickly, how to multiply two numbers easily,how to multiply two numbers fastly , fast way to multiply two digit numbers, fastest way to multiply two digit numbers any number however large with 11 in fraction of seconds. With the help of this method , we can multiply any number in just 2 - 3 seconds.

1 Place zero at right side of the multiplicand.

2 Keep on adding each digit of the multiplicand from extreme right to its neighbour till end , if any stage sum is obtained greater than 10 then carry of 1 will be added to next step.

Let us understand it better with the help of video

• Examples 

52324 × 11= ?

1 Place zero at right end of the multiplicand like this 523240

2 Now add 0 to its neighbour 4 as 0 + 4 = 4

3 Now add 4 to its neighbour 2 as 4 + 2 = 6

4 Now add 2 to its neighbour 3 as 2 + 3 = 5

5 Now add 3 to its neighbour 2 as 3 + 2 = 5

6 Now add 2 to its neighbour 5 as 2 + 5 = 7

7 Place left most digit as it is = 5

8 Write the digits so obtained  from top to bottom as right to left

So Answer will be 5,75,564

This video demonstrate the process of multiplication very easily

• Examples 

4543423 × 11= ?

1 Place zero at right end of the multiplicand like this 45434230


2 Now add 0 to its left neighbour 3 as 0 + 3 = 3


3 Now add 3 to its left neighbour 2 as 3 + 2 = 5


4 Now add 2 to its left neighbour 4 as 2 + 4 = 6


5 Now add 4 to its left neighbour 3 as 4 + 3 = 7


6 Now add 3 to its left neighbour 4 as 3 + 4 = 7


6 Now add 4 to its left neighbour 5 as 4+ 5 = 9


6 Now add 5 to its left neighbour 4 as 5 + 4 = 9


7 Place left most digit as it is = 4

8 Write the digits so obtained  from top to bottom as right to left

So Answer will be 4,99,77653

3598678 × 11= ?

1 Place zero at right end of the multiplicand like this 3598678

2 Now add 0 to its neighbour 8 as 0+8= 8

3 Now add 8 to its neighbour 7 as 8+7=15 write 5 and carry 1 to next step

4 Now add 7 to its neighbour 6 as 7+6=13+1 (carry)=14 write 4 and carry 1 to next step

5 Now add 6 to its neighbour 8 as 6+8 = 14+1(carry) = 15 write 5 and carry 1 to next step

6 Now add 8 to its neighbour 9 as 8+9=17+1(carry) =18 write 8 and carry 1 to next step

7 Now add 9 to its neighbour 5 as 9+5=14+1(carry) = 15 write 5 and carry 1 to next step

8 Now add 5 to its neighbour 3 as 5+3=8+1(carry) = 9

9 Place left most digit as it is = 3

11 Write the digits so obtained from top to bottom as right to left


So Answer will be 3,95,85458

8923586 × 11 = ?

1 Place zero at right end of the multiplicand like this 89235860

2 Now add 0 to its neighbour 6 as 0+6 = 6

3 Now add 6 to its neighbour 8 as 6+8=14

write 4 and carry over 1 to next step

4 Now add 8 to its neighbour 5 as 8+5=13+1(carry) = 14 

write 4 and carry over 1 to next step

5 Now add 5 to its neighbour 3 as 5+3=8+(1)carry= 9

6 Now add 3 to its neighbour 2 as 3 + 2 = 5

7 Now add 2 to its neighbour 9 as 2+9=11 

write 1 and carry over 1 to next step

8 Now add 9 to its neighbour 8 as 9+8=17+(1)carry = 18 

write 8 and carry over 1 to next step

9 Now add 1(carry ) to its neighbour 8 as 1+8 = 9

Write all  the digits so obtained  from top to bottom as right to left.

So answer will be 98,159,446

35681237 ×11 = ?

1 Place right most digit 7 of multiplicand as right most digit of answer.

2 Keep on adding right sided digit to its left sided digit in pairwise.

3 If the sum at any time is found to be more than 10, then take "1" as carry over to next step every time.

4 Repeat the process till last digit. So After 1st step we shall have 7

After 2nd step we shall have 7+3=10 =0 (right sided digit of 10 ) and 1 as carry to next step.

After 3rd step we shall have 3+2=5+1=6 and no number as carry to next step.

After 4th step we shall have 2+1=3 and no number as carry to next step.


Similarly we get 1+8=9,


and 8 + 6 = 14 = 4 as (right sided digit of 14 ) and 1 as carry to next step.


5 + 6 = 11 + 1 = 12 = 2 (right sided digit of 12), and


3+8 = 8 + 1 = 9;

And the last digit = 3

Now write all the highlighted digits from bottom to top .

So Answer will be    392493067

These are some of the examples demonstrated in the video given below

Application of this Method

If we have to multiply 666854×55
then rewrite given product as 666854×(11×5)
Now multiply 666854×11 as follows
Step 1

Place right most digit 4 as result and keep on adding the digits to its left one by one which gives 7335394, and
Step 2

Now place 0 as right most digit of this result i. e.73353940 ,


Step 3 

Now divide with 2 we get 36676970 and this is the Final answer.
Example

Let us multiply 35987604 × 55

Rewrite 35987604 × (11×5)

Multiply 35987604 × 11 = 395863644

Now place "0" at extreme right of this number it become 3958636440 , 

Now divide this number with 2 to get the Answer 1979318220.
Now Fast multiplication with one more Example

69852364639×55

Step 1 

 1st multiply the given number with 11 by placing and adding digits from left to right 9,12,9,10,10,9,5,7,13,17,15,6 (if total greater than 10 ,carry 1 to next number) like this 9,2,0,1,1,0,6,7,3,8,6,7.

Step 2

 Write these numbers from left to right, place zero at end and divide by 2 to get the answer like this  7683760110290 → 384,188,005,5145.

Conclusion

This shortcuts method was to multiply a number with 11 . Thanks for spending your precious time to read this post ,If you liked this post . Please share it with your friends and also follow me on my blog to encourage me to do better than best. See your in next post, till then Bye.....

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HOW TO FIND THE TRANSPOSE OF MATRIX WITH AN EASY METHOD

In this post we are going to learn "what is transpose of matrix, calculate  transpose ,define transpose, meaning of transpose " .  As it is clear from its name  transpose means trans + pose i. e. transfer + position ( transfer of position ) ,  the transpose   of any matrix is obtained by transfer of Rows into Columns And vice versa. i.e transforming 1st row to 1st column and transforming 2nd row to 2nd column and so on for 3rd ,4th and 5th rows and columns. This is very easy and interesting topic in matrices and determinants.

 To find the transpose of the matrix 

1^st of all  shift all the elements which are in 1st row to 1^st column as

5

5

2
,then shift the elements which are in      2nd row to 2^nd column as  

-1

-3

 7

similarly  shift all the elements which are in 3rd row to 3^rd column as

4

2

8

And the matrix so obtained is the transpose matrix. We can  check that same colour row have been  transformed to same colour Column
        

Now we shall take one  example to find the transpose matrix

Here Given matrix have 3 rows and 4 columns .It means we shall have 4 rows and 3  columns  in transpose matrix.

1st shift all the elements which are in 1st row to 1^st column as

5
8               
6
-4

shift all the elements which are in 2nd row to 2nd column as

3
5
8
3

shift all the elements which are in 3rd row to 3rd column as 

-3
8
-7
6

so shifting the corresponding Rows into Corresponding columns. We can  check that same colour rows are transformed to same colour columns.

Now we shall take one more  example to find the transpose of matrix .Here Given matrix have 3 rows and 4 columns,It means  we shall have 4 rows and  3 columns in transpose of that matrix.

1st shift all the elements which are in 1st row to 1^st column as

 7

-1

-2

 5

shift all the elements which are in 2nd row to 2nd column as

4

4

3

3

shift all the elements which are in 3rd row to 3rd column as 

  8

 9

 6

-1

so shifting the corresponding Rows into Corresponding columns. We can that check same colour rows are transformed to same colour column.

Let us take an example  Where  A =

Step 1 Then  on transforming 1st Column  to 1st Row  ,we have 

4             -3         9          as 1st  Row

Step 2 Then  on transforming  2nd Column  to 2nd Row  ,we have 

5             2          -2          as  2nd Row

||ly    on transforming   3rd  Column  to 3rd Row  ,we have 

7          3          8             as 3rd Row

After  taking Transpose  A' will be  

Let us take more example to find out the transpose of matrices. These are two examples , both of which are of  3×3 orders. Hence the transpose of these matrices will be  again 3×3. 


1st consider matrix G , after transforming its 1st row into column , the 1st column of the transpose matrix of  G '  will be  -2,  -5,  4.
After transforming its 2nd row into column , the 2nd column of the transpose matrix of  G '  will be   -5,  7 ,  3. And after transforming its 3rd row into column , the 3rd  column of the transpose matrix of  G '  will be  4,  3,  8. If we write the transpose of  matrix G then we can see that there is no difference between the matrix G and the transpose of the matrix G.

Now consider matrix H , after transforming its 1st row into column , the 1st column of the transpose matrix of  G '  will be  2,  3,  4.      
After transforming its 2nd row into column , the 2nd column of the transpose matrix of  H '  will be   3, 5 ,  6. And after transforming its 3rd row into column , the 3rd  column of the transpose matrix of  H '  will be  4,  6,  7. Again in the case of matrix H , Matrix H and its Transpose matrix   H '  are same.

Here is more interesting Example of Transpose of this Matrix,This matrix have 3 Rows and 3 columns,after taking Transpose this matrix still have 3 Rows and 3 columns,

Now Take Transpose of this Matrix 

what ?

Surprise to see that the Transpose of some of the Matrices are the Matrices Itself, i. e. if A¹= A

Such Matrices are called Symmetric  Matrices.

Again if we take the transpose of the matrix given below and take -1 common from the matrix so obtained i .e  this matrix will be equal to transpose of the negative of the transpose of the given matrix

then   A¹ =  - A ,  Such Matrices are called  Skew Matrices.

Final words

This post was regarding what is transpose of matrix, calculate  transpose , define transpose,meaning of transpose , If you learn  something from this post then share it  with your friends  and also follow me on my blog ,We  shall meet  again in next post , till then  Good Bye ..................................

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SHORTCUT METHOD TO FIND THE SQUARE OF TWO AND THREE DIGITS NUMBERS ,MULTIPLICATION TRICKS

We shall learn a short cut method to find the square of two and three digits numbers . This method is very much time shaving and easy.

Square of Two And Three Digits Numbers

In order to save time  in computing long arithmetic  calculations ,sometime we need short cuts which are effective to save time and easy to understand. There are many shortcuts available to compute faster ,one of them is discussed below with the help of Examples .

Example :   To Find the Square of Two Digits Numbers Ending with 5

Here I shall show  how square of 35 can be  found quickly.

Step 1 

when unit/(Right most)  place of a number ends with 5 
then place 25 as last two digits (i. e Right most ) of the result.

Step 2 

Take the digit which is on the unit's place and multiply it with its successor (next digit from it), Here in this case the 1st two digits are 12, the product of 3 and its  successor 4. 

Step 3

The result will be  four digit numbers whose first two digits are answer  of step 2 and next two digits are answer of step 1

The square of  35 = (3*4)(25)
                                        = 1225

Example

To find Square of   65

1  Place 25 as last two digits of the answer as given number ends with 5

2 Place the product of  6 and it successor i.e. 7 which is equal to 42

So Answer will be   4225

Hence
Square of 55 is (5*6)  = 30  as 1st two digits (25) as last two digits = 3025

Square of 75 is (7*8)  = 56  as 1st two digits(25) as last two digits = 5625

Square of 45 is (4*5) =  20 as 1st two digits (20) as last two digits = 2025

Square of 25 is (2*3) = 06 as 1st two digits (25) as last two digits = 625

Square of 85 is (8*9) = 72 as 1st two digits (25) as last two digits = 7225

2nd Example : To find the Square of  two digits number not ending with 5

Here  we are taking example of  78,  the digit in unit's place is 8 and the digit in ten's place is 8, result of square of any two digits number will be written in four digits .

               -          -         -        -
         1st     2nd      3rd    4th

Step 1 

Square  unit  place's digit if it is two digits number then write units place digits in 4th place and reserve the ten place 's digit for next step , Here square of 8 is 64 so write 4 at 4th digits in answer and reserve 6 as carry forward  for next step.

8*8 = 64, 4 as 4th place  and 6 as carry forward

Step 2

          Now multiply both the digits of our example and then again multiply it with 2 and add carry forwarded number of the step 1 to it, write the extreme right digit of the result so obtained in place of 3rd place, and reserve the remaining digit/digits as carry forward for next step.

Here 2*7*8=112 + 6 = 118 , 8 at 3rd place and 11 as carry forward for next step.

Step 3

        Take the square of  ten's place digit and add  the carry forwarded result from previous step and place on 1st and 2nd place.

 Here 7*7=49+11=60, 6 at 1st digit and 0 at 2nd digit .

The square of 78 is 6084

3rd Example : To find the Square of  two digits number not ending with 5

Here  we are taking example of  86, here in unit's place is 6 and in ten's place is 8,result of square of any two digits number will be written in four digits .

               -          -         -        -
         1st     2nd      3rd    4th

Step 1 

         Take the Square of  unit  place's digit if it is two digits number then write units place digits in 4th place and reserve the ten place 's digit for next step , Here square of 6 is 36 so write 6 at 4th digits in answer and reserve 3 as carry forward  for next step.

6*6=36, 6 as 4th place  and 3 as carry forward

Step 2

          Now multiply both the digits of our example and then again multiply it with 2 and add carry forwarded digit of the step 1 to it, write the extreme right digit of the result so obtained in place of 3rd place, and reserve the remaining digit/digits as carry forward for next step.

Here 2*8*6=96+3=96 , 6 at 3rd place and 9 as carry forward for next step.

Step 3

        Take the square of  ten's place digit and add  the carry forwarded result from previous step and place on 1st and 2nd place.
 Here 8*8=64+9=73, 7 at 1st digit and 3 at 2nd digit

Step 4

The square of 86 is 7396

To learn one more maths tricks

Example :   To Find the Square of Three Digits Numbers Ending with 5

Split this three digits number into two number , right extreme number as unit place and remaining two digits as Ten's place (assumed as one digit).

To find square of 135

step 1 

when unit place of a number ends with 5 
then place 25 as last two digits (i. e right most ) of the result

Step 2 

Take the digits which is on the unit's place and multiply it with its successor (next digit from it), here in this case the  product of 13 and its  successor 14 is equal to 182

Step 3

The result will be  five digit numbers whose three digits are answer  of step 2 and next two digits are answer of step 1

The square of 135 = (13*14)(25) = 18225

2nd Example : To Find the Square of Three Digits Number Not Ending with 5

Now we are taking example of  132, here  unit's place is 2 and  ten's place is 13 (only assumption) ,result of square of any two digits number will be written in five digits .

               -          -         -        -     -
         1st     2nd      3rd    4th   5th

Step 1 

        Find  Square of  unit place's digit if it is two digits number then write units place digits in 4th place and reserve the ten place 's digit for next step , So  square of 2 is 4 therefore  write 4 at 5th digits in answer and reserve 0 as carry forward  for next step.

2*2=4, 4 as 5th place  and 0 as carry forward

Step 2

          Now multiply both the remaining digits of our example and then again multiply it with 2 and add carry forwarded digit of the step 1 to it, write the extreme right digit of the result so obtained in place of 4th place, and reserve the remaining digit/digits as carry forward for next step.

Here 2*13*2=52+0=52 , 2 at 4th  place and 5 as carry forward for next step.

Step 3

        Take the square of  ten's place digit (assumed 13) and add  the carry forwarded result from previous step and place on 1st , 2nd and 3rd  places.

Here 13*13=169+5=174, 1 as 1st digit , 7 as 2nd digit and 4 as 3rd digit  

step 4

The square of   132 is   17424.



           

3rd Example : To Find the Square of Three Digits Number Not Ending with 5

Here  we are taking example of  146, here  unit's place is 2 and  ten's place is 14 (only assumption) ,result of square of any two digits number will be written in four digits .

               -          -         -        -     -
         1st     2nd      3rd    4th   5th

Step 1 

Find  Square  of unit  place's digit if it is two digits number then write units place digits in 4th place and reserve the ten place 's digit for next step , Here square of 6 is 36 so write 6 at 5th digits in answer and reserve 3 as carry forward  for next step.

6*6 = 36, 6 as 5th place  and 3 as carry forward

Step 2

          Now multiply both the remaining digits of our example and then again multiply it with 2 and add carry forwarded digit of the step 1 to it, write the extreme right digit of the result so obtained in place of 4th place, and reserve the remaining digit/digits as carry forward for next step.

Here 2*14*6 = 168 + 3 = 171 ,   1 at 4th  place and 17 as carry forward for next step.

Step 3

        Take the square of  ten's place digit (assumed 13) and add  the carry forwarded result from previous step and place on 1st ,2nd and 3rd  places.

 Therefore  14*14 = 196 + 17=213, 2 as 1st digit , 1 as 2nd digit and 3 as 3rd digit  

step 4

The square of   146 is   21316

To Find The square of the Number 127

Step 1   Square of 7 =49 write 9 at unit place and carry 4 to next step.

Step 2    Multiply 2*12*7 =168 , Add carry of 4 to it and get  168 + 4 = 172, write 2  at hundred place and carry 17 to last step

Step 3 Add 17 to the  Square of 12 i.e  144  like this 144+17  = 161,
write 161 as 1st three digits of the answer
So the Answer is 16129

To Find The square of the Number 257

Step 1   Square of 7 = 49 write 9 at unit place and carry 4 to next step.

Step 2    Multiply 2*25*7  = 350 , Add carry of 4 to it and get  350 + 4 = 354 , write 4  at hundred place and carry 35 to last step

Step 3 Add 35 to the  Square of 25 i.e  625  like this 625 + 35  = 660,

write 660 as 1st three digits of the answer


So the Answer is 66049

                                        

Final Words

Thanks for spending your precious time to this post of multiplication tricks, finding square of two and three digits numbers quickly , If you really gained something from this post ,then  Share this post with your near and dear , Also follow me on my blog . We shall meet again in next post , till then Good Bye.

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