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Squaring Techniques

 

Squaring a number which ends in 5

  • Last two digits of the square are always 25.
  • To find the number which comes before 25, just perform the operation n x (n+1), where n is the digit before 5 in the original number.
  • By putting the number received in step 2, just before 25 and you get your square.

 

 Example
Find the Square of 65.
Solution

We can find the same in the following 2 steps –

  1. Last two digits are 25.
  2. The digit before 5 is 6, if we perform n x (n+1) operation on this, then we will be 6 x (6+1) = 42
  3. Hence the square of 65, will be 4225.

Finding Square of any two digit numbers
 

The method is to use (a + b)2 formula in the following manner a2 / 2ab / b2.

 

Examples

Find the square of 85.

Solution

we can break 85 into two parts a (8) and b (5) and use the formula-


To find 872 you are required to do the following:

 

Finding Square of a number which is between 25 – 50
 

  • Assume, we have to find the square of N. First find the difference with 50 (lets say the difference is D)
  • Find the square of D. It will give the last 2 digits of square.
  • Subtract D from 25, ie find (25-D). It will give us the first two digit of square.

 

Example

Find 432.

Solution

Steps-

  1. Find out how far is the number from 50. Eg: 43 is 7 units far from 50
  2. Square the above number. The result will be the last two digits of the required result i.e., 72 = 49, which is the last two digits of the required result. __49.
  3. Subtract the initial difference of step 1 from 25 i.e.,25 – 7 = 18, this will be the first two digits of the result. 18__.
  4. So, 432 = 1849. 

Finding Square of a number which is between 51 - 75

  1. Assume, we have to find the square of N. First find the difference with 50 (lets say the difference is D)
  2. Find the square of D. It will give the last 2 digits of
  3. Add D in 25, ie find (25+D). It will give us the first two digit of square.

 

Example

Find 522.

Solution

Steps:

  • Find out how far is the number from 50. Eg: 52 is 2 units far from 50.
  • Square the above number. The result will be the last two digits of the required result. i.e., 22 = 04, which is the last two digits of the required result. __04.
  • Add the initial difference of step 1 to 25 i.e., 25 + 2 = 27, this will be the first two digits of the result. 27__.
  • Hence, Answer is 2704.

Finding Square of a number which is between 75 – 100

 

  • Assume, we have to find the square of N. First find the difference with 100 (lets say the difference is D)
  • Find the square of D. It will give the last 2 digits of square.
  • Find the difference between the number (N) and Difference (D) i.e. find (N-D). It will give us the first two digit of square.

 

Example

Find the square of 93?

Solution

The deficiency of 93 is 7 because it is 7 less than 100 (base). Now, 93 – 7 = 86. This gives us the left-hand part of the answer. The square of the deficiency is 7 × 7 = 49, and this is the right – hand part of the answer. The working can be done mentally and the answer, 8649, written straight down.

Finding Square of a number which is between 100 and 125
 

  • Assume, we have to find the square of N. First find the difference with 100 (lets say the difference is D)
  • Find the square of D. It will give the last 2 digits of square.
  • Add the number (N) and Difference (D) ie find (N+D). It will give us the first two digit of square.

 

Example

Find the square of 103?

Solution

The surplus is 3. Increase 103 by 3, gives 106. This gives the left-hand part. The square of the surplus is 3 × 3 = 09. And this is the right-hand part of the answer. The base has two zeros and so the right-hand part is 09 and not just 9. So, the answer is 10609. 
 

Find the Square of number containing Repeated 1

Step-1: Count the Digit, Count = n

Step-2: Now starting from 1 write the number till n

Step-3: Now starting from n write number till 1

 

Example

Find the square of 1111.

Solution

First we see that there are 4 times 1.
Now we write number from 1 to 4.
Now again from 4 to 1
So our answer is 1234321. 

 

Example

Find the square of 111.

Solution
First we see that there are 3 times 1.
Now we write number from 1 to 3.
Now again from 3 to 1
So our answer is 12321.  
 

Find the square of the number containing Repeated  9

Step-1: Count the digit, Count = n

Step-2: First write (n - 1) times 9 then 8

Step-3: Again (n - 1) times 0 then 1

 

Example

Find the square of 9999.

Solution
We see that there are 4 times 9
Now we write (4 - 1 = 3) times 9 then 8, {9998}
Now 3 times 0 then 1{0001}
So Answer Will be 99980001
 

 

Example

Find the square of 999999.

Solution

We see that there are 6 times 9
Now we write (6 - 1 = 5) times 9 then 8, {999998}
Now 5 times 0 then 1{000001}
So Answer Will be 999998000001
 

Find the square of the number containing Repeated  3

Step-1: Count the digit, Count=n

Step-2: First write (n-1) times 1 then 0

Step-3: Again (n-1) times 8 then 3

 

Example

Find the square of 333333.

Solution

We see that there are 6 times 3
Now we write (6 - 1 = 5) times 1 then 0, {111110}
Now 5 times 8 then 9{888889}
So Answer Will be 111110888889





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