Square Of Number (Fastest Approach)

Number containing 5 at the end:

Getting square of a number ending in 5 is simple.If the last digit contain 5, the last 2 digit of square will be 25 ,whatever the earlier part of number multiply it with one more than itself.

For example: 35²=(3*4)|25
45²=(4*5)|25

Number containing 1 and 6 at the end:
First observe the following:

1²=1=1
2²=4=1+3
3²=9=1+3+5
4²=16=1+3+5+7
5²=25=1+3+5+7+9

Above shows that to get n² ,we add upto first n odd number. for example :

16²=sum of first 16 odd number

There from above we conclude that:

26²=25²+26th odd number i,e 625+51=676

31²=900+31st odd number i,e 900+61=961

Number containing 4 and 9 at the end:

Let's take example:

29²=30²- 30th odd number

34²=35²- 35th odd number

Number containing 7 at the end:

27²=26²+27 odd nomber= 25²+26th odd number+27th odd number

The sum of the 26th odd number and 27th odd number is same as 4 times 26. Hence,

27²=25²+4*26 =625+104 =729

77²=5625+304(4 times 76) = 5929


Number containing 3 at the end:

53²= 55²-4 times 76 = 3025-216=2809

93²=9025- 376=8649