Base 150 : i.e for numbers between 126-175.
1. Get the difference between 150 and given number and square diff(Write it on right side such that it will consume last two digits and carry 100th digit.
2. Subtract the difference( if number < 150) or Add the difference (if number > 150) in given number and multiple the number by 1.5. And Write result in front of result of case 1.

e.g: 1. 160×160
Case1- diff- 10-square- 100
Add diff- 160+10=170
Multiply by 1.5- 170×1.5=255
Concate 255 and 100 carry 100th position
Ans: 25600

E.g 136×136
Diff- 14- square - 196
Substarct 14 from 136= 122
122×1.5 = 183
Concate 183 and 196 such that 1 will carry
Ans: 18496

Understand the difference when you find it with calculations and by TRICK given above.

Base 200 : i.e for numbers between 175-225.
1. Get the difference between 200 and given number and square diff(Write it on right side such that it will consume last two digits and carry 100th digit.
2. Subtract the difference( if number < 200) or Add the difference (if number > 200) in given number and multiple the number by 12. And Write result in front of result of case 1.

e.g: 1. 212
Case1- diff- 12-square- 144
Add diff- 212+12=224
Multiply by 2: 224×2=448
Concate 448 and 144 carry 100th position
Ans: 44944

E.g 176×176
Diff- 24- square - 576
Substarct 24 from 176 = 152
152×2 = 304
Concate 304 and 576 such that 5 will carry
Ans: 30976

I think you can now easily find the squares of any number by understanding the above shortcuts. From tomorrow onwards, we will start learning how to identify whether a number is a perfect square or not, along with the rules for it.
Excited?

Here, we find square of number 212 which is 44944.
This form of number are called palindrome that is from middle they are at same distance on both the sides and similar also.

More examples of Palindrome are:

• 1^2 = 1
• 11^2 = 121
• 111^2 = 12321
• 1111^2 = 1234321
• 11111^2 = 123454321
•
•
And so on upto 9 times 1.

How to identify given number is perfect square or not?

• Last digit of a perfect square is NEVER 2,3,7 or 8 And it's always 0,1,4,5,6 and 9.
e.g: 100, 81, 64, 25, 36, 529...

• If last digit of square is 5, then the 2nd last digit is always 2.
e.g: 25, 625, 1225...

• If last digit of square is 6, then the 2nd last digit is always ODD.
e.g: 36, 256...

• If last digit of a square is 1,4 and 9 then 2nd last digit is always EVEN.
e.g: 81, 121, 144, 484, 49, 1089...

• The number of zero's at the end of a square is always EVEN & the non zero part should be a perfect Square.
e.g: 900, 25600, 10000...

• 7744 is the Only Perfect square number of the form/type XXYY.
i.e 88^2 = 7744.

Cube And Cube Roots:
You should know the Cubes of numbers from 1-10.
1^3- 1
2^3- 8
3^3- 27
4^3- 64
5^3- 125
6^3- 216
7^3- 343
8^3- 512
9^3- 729
10^3- 1000

You can easily find cube of any number by using formulas given:

(a + b)^3 = a^3 + 3a^2b + 3ab^2+ b^3
(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3

Example:

23^3 = (20 + 3)³

Step 1 — Apply Formula

(20 + 3)^3 = 20^3 + 3(20^2×3) + 3(20×3^2) + 3^3

Step 2 — Solve
= 8000 + 3(400×3) + 3(20×9) + 27
= 8000 + 3600 + 540 + 27 
= 12,167

7³ = (10 − 3)³

Step 1 — Apply Formula

(10 - 3)^3 = 10^3 - 3(10^2×3) + 3(10×3^2) - 3^3

Step 2 — Solve

= 1000 - 3(100×3) + 3(10×9) - 27
= 1000 - 900 + 270 - 27 
= 343

How to identify cube of number: There is trick:
Check last digit of number and Check last digit of answer also

Number Cube Last digit
1 1 1
2 8 8
3 27 7
4 64 4
5 125 5
6 216 6
7 343 3
8 512 2
9 729 9
10 1000 0

By checking only last digits you can eliminate at least 1 or 2 or 3 options.
You will be close to answer.

Here it is expected that you should check the last digit...
As 4 is there at last in 54... According to shortcut...
There should be 4×4×4 = 16×4 i.e. 6×4= 24 i.e. 4 should be there at last digit in answer.

As there are two such options.
We will go for formula....

54³= (50+4)³
=12500+3×2500×4+3×50×16+64
=125000+30000+2400+64
=1,57,464

LCM (Least Common Multiple)

Definition →The smallest number that is divisible by all given numbers.

Steps to Find LCM
Method 1 – Prime Factorization

• Factorize each number into prime numbers.

• Take each prime factor only once with its highest power.

• Multiply them together.

Example:
Find LCM of 12 and 18.

12 = 2² × 3¹
18 = 2¹ × 3²

LCM = 2² × 3² = 36

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Method 2 – Division Method (Shortcut)

• Divide the numbers by prime numbers till all become 1.

• Multiply all divisors together.

Example:
Find LCM of 15, 20, 30.

Divisor: 15 , 20, 30

2- 15 10 15
2- 15 5 15
3- 5 5 5
5- 1 1 1

LCM = 2 × 2 × 3 × 5 = 60