**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.

We can find the same in the following 2 steps â€“

- Last two digits are 25.
- 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
- 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 **a ^{2 }/ 2ab / b^{2.}**

** **

**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 87^{2} you are required to do the following:

**Finding Square of a number which is between 25 â€“ 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.*

Find 43^{2.}

**Steps-**

- Find out how far is the number from 50. Eg: 43 is 7 units far from 50
- 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.
- Subtract the initial difference of step 1 from 25 i.e.,25 â€“ 7 = 18, this will be the first two digits of the result. 18__.
- So, 43
^{2}= 1849.

**Finding Square of a number which is between 51 - 75**

*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**Add D in 25, ie find (25+D). It will give us the first two digit of square.*

Find 52^{2}.

**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., 2
^{2}= 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 â€“ 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.*

Find the square of 93?

The deficiency of 93 is 7 because it is 7 less than 100 (base). Now, 93 â€“ 7 = 86. This gives us the left-hand part of the answer. The square of the deficiency is 7 Ã— 7 = 49, and this is the right â€“ 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.*

Find the square of 103?

The surplus is 3. Increase 103 by 3, gives 106. This gives the left-hand part. The square of the surplus is 3 Ã— 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

Find the square of 1111.

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**

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

Find the square of 9999.

**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

Find the square of 333333.

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