Properties of Square number

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

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:

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

its square ends with 6

Example:

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