What are the Properties of Perfect Square Numbers

Learning the properties of perfect square numbers is important to check if a number is a perfect square or to guess what would be the digit at the unit place of the square. Understanding these properties is vital to learn some shortcuts that are used to solve questions based on squares. There are many interesting properties and patterns of the square numbers and knowing them help build a solid foundation for understanding various math concepts such as Pythagorean triplets. These properties are also applied in solving algebraic identities covered under various topics of mathematics. Let's explore more about square numbers and their various properties one by one in detail.

Table of Contents

• List of Properties of Perfect Squares
• Solved Examples Based on Properties of Perfect Squares
• Frequently Asked Questions of Properties of Perfect Squares

List of Properties of Perfect Squares

Property 1: 

Perfect square numbers always end with 0, 1, 4, 5, 6 or 9 at their units place.

Example: 100, 81, 64, 25, 49, 256, 169 etc. are perfect squares

But not all numbers ending with 0, 1, 4, 5, 6 or 9 at their units place are perfect squares. Example: 20, 31, 54, 45, 56, 39, etc. are not perfect squares.

Property 2: 

A perfect square number never has 2, 3, 7 or 8 at its units place.

Example: 16, 81, 36, etc. are perfect squares whereas 12, 23, 37, 48, etc are some examples of numbers that are not perfect squares as they end in 2, 3, 7 and 8 respectively.  

Property 3: 

• When the units place digit of a number is 1 (one) or 9 (nine), then the square of that number will have 1 at its units place.
  Example:$9^2 = 81$
• When the units place digit of a number is 4 (four) or 6 (six), then the square of that number will have 6 at its units place.
  Example:$16^2 = 256$
• When the units place digit of a number is 3 (three) or 7 (seven), then the square of that number will have 9 at its units place.
  Example:$27^2 = 729$
• When the units place digit of a number is 2 (two) or 8 (eight), then the square of that number will have 4 at its units place.
  Example:$18^2 = 324$
• When the units place digit of a number is 0 (zero), then the square of that number will also have 0 at its units place.
  Example:$20^2 = 400$

Property 4: 

A number that ends with zero will have a square that has 2n zeros at its end. That is, the number of 0s at the end of a perfect square is always even.

Example:$100^2 = 10000$ Here 100 has two zeros; its square has 2 × 2 = 4 zeros. 

Property 5: 

An even number will always have an even square, i.e., when an even number is multiplied by itself, the result is always an even number.
Example:  $6^{2}$  = 36

Property 6: 

An odd number will have a square that is an odd number, i.e., when an odd number is multiplied by itself, the result is always an odd number.
Example:   $5^{2}$ = 25

Property 7: 

The square of an n-digit number has 2n or (2n‒1) digits in its square.
Example: 3² = 9, 11² = 121; here the 1-digit number has (2n - 1) = (2 × 1 - 1) = 1 digit in its square. 11 is a two-digit number having (2 × 3 - 1) = 3 digits in its square.

Solved Examples Based on Properties of Perfect Squares

Example 1: Without calculating, determine whether 7928 can be a perfect square.
Solution: Given Number = 7928
Digit at units place = 8
Property: A perfect square can only end in 0, 1, 4, 5, 6, or 9.
Since 7928 ends in 8, it is NOT a perfect square.
Hence, 7928 cannot be a perfect square.

Example 2: Without calculating, determine if 47² is odd or even.
Solution: As per the property of square of an odd number, the square of an odd number is always odd.
Therefore 47² is odd. 
47² = 2209 

Example 3: Check whether 625 is a perfect square.
Solution: The last digit of 625 is 5. Numbers ending in 5 can be perfect squares. 25 × 25 = 625. Hence 625 is a perfect square.

Example 4: Is 245 a perfect square?
Solution: A perfect square never ends with 2, 3, 7, or 8. Since 245 ends with 5, it may be a perfect square.
 $15^ 2 = 225$ and  $16^2 = 256$. Since 245 lies between 225 and 256, it is not a perfect square.