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)2 +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
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
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)2 +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