Let’s look at the square of numbers from 1 to 50

Number Square
1 1
2 4
3 9
4 16
5 25
6 36
7 49
8 64
9 81
10 100
11 121
12 144
13 169
14 196
15 225
16 256
17 289
18 324
19 361
20 400
21 441
22 484
23 529
24 576
25 625
30 900
35 1225
40 1600
45 2025
50 2500

Let’s see some pattern in it, and find properties of square numbers

 

A Square number can only end with digits 0, 1, 4, 5, 6, 9

Example :

1, 4, 9, 16, 25, 36, 49, 64, 81, 100 are square numbers.

But, 28, 97 will not be a perfect square

 

Number of zeroes at the end of a perfect square is always even

Example :

2500 is a perfect square

100 is a perfect square

But, 80 is not a perfect square

4000 is also not a perfect square

 

 

Square of even numbers are always even,

Square of odd numbers are always odd

Example :

Square of 2 is 4,

Square of 6 is 36

And

Square of 7 is 49

Square of 9 is 81

 

If a number has 1 or 9 in its unit place,

its square ends with 1

Example:

Number

Square

1

1

9

181

11

121

19

361

 

If a number has 4 or 6 in its unit place,

its square ends with 6

Example:

Number

Square

4

16

6

36

14

196

16

256

 

Unit digit of square of any number will be the unit digit of the square of its last digit

Example:

For number 29

Unit digit of 29 2 = Unit digit of 9 2

     = Unit digit of 81

    = 1

 

For number 76

Unit digit of 76 2 = Unit digit of 6 2

   = Unit digit of 36

  = 6

 

A perfect square always leaves remainder 0 or 1 when divided by 4

Perfect Square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100,..

 

  • 1 on divided by 4 leaves remainder 1
  • 4 on divided by 4 leaves remainder 1
  • 9 on divided by 4 leaves remainder 0
  • 16 on divided by 4 leaves remainder 1
  • 25 on divided by 4 leaves remainder 1
  • 36 on divided by 4 leaves remainder 0

 

So, if we need to check whether 98 is a perfect square,

we check its remainder when divided by 4

98 on divided by 4 leaves remainder 2

So, it is not a perfect square

 

Square number is sum of consecutive odd numbers

Check explanation here

 

For any Natural number m greater than 1 (2m, m 2 - 1, m 2 +1) is a pythagoras triplet

Check explanation here

 

 

  1. Chapter 6 Class 8 Squares and Square Roots
  2. Concept wise

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.