Square and Square roots


Squares of Number having 5 in the units 

Without actual multiplication, find the squares of the following :-

a. 75 
We find out the square of 75 in the following way :
75 2  =  7 x 8  25 = 5625

Squares of Numbers Ending in 1. 
a. 91
we find out the squares of 91 in the following way:
912 =  
9 x 92 1 = 8281

Pythagorean Triplets
Three natural numbers a, b, c are said to form a Pythagorean Triplets if a2 + b2 = c2 . Also for every natural number m >1, (2m , m2 - 1, m2 +1).

1. Find the Pythogorean triplets whose smallest number is 18.
If 2m, m2 - 1, m2 +1 form a Pythagorean triplet, then 2m is the smallest number of this triplets 
2m = 18.
m = 9.
now, m2 - 1 = (9)2 - 1 = 81 - 1 = 80 
m2 +1 = (9) +1 = 81 +1 = 82
Ans. Three triplets are 18, 80, 82 

Q 1. Find the least square number which is divisible by 6, 9 and 15 .
Solution :- The LCM of 6,9 and 15 is 90.
90 is the least number divisible by 6, 9 and 15. 
By Prime factorisation, we get
90 = 2 x 3 x 3 x 5 

To make it a perfect square, it must be multiplied by (2 x 5) = 10.
now, 90 x 10 = 900
Ans :- 900 is the smallest perfect square divisible by 6, 9 and 15.

Units digit of a square and its square root 

 
Units Digit of a Square  Units Digit of a Square Root
0 0
1 1 or 9
4 2 or 8
5 5
6 4 or 6 
9 3 or 7

Methods of Finding the Square root of perfect Squares 

The Square root of a perfect square may be found be any of the following two methods 
1. Prime factorisation method 
2. Long division method 

 
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