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 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

### its square ends with 1

Example:

 Number Square 1 1 9 181 11 121 19 361

### 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
3. Properties of square numbers

Properties of square numbers 