Properties of Square Numbers are explained with examples here. The square numbers are the numbers that are produced when a number is multiplied to itself (Check: Square Numbers). For example, 4 is a square number that is produced when 2 is multiplied by itself. It is expressed as:
2^{2} = 4 [Two squared equals four]
In this article, we will discuss the different properties of square numbers. But before we proceed with the properties, let’s have a look at square numbers from 1 to 50.
Square Numbers 1 to 50
Here is the list of square numbers from 1 to 50, where N is the natural number and N^{2} is the square of N. The list will help us to learn about the properties of square numbers.
N 
N^{2} 
N 
N^{2} 
N 
N^{2} 
N 
N^{2} 
N 
N^{2} 
1 
1 
11 
121 
21 
441 
31 
961 
41 
1681 
2 
4 
12 
144 
22 
484 
32 
1024 
42 
1764 
3 
9 
13 
169 
23 
529 
33 
1089 
43 
1849 
4 
16 
14 
196 
24 
576 
34 
1156 
44 
1936 
5 
25 
15 
225 
25 
625 
35 
1225 
45 
2025 
6 
36 
16 
256 
26 
676 
36 
1296 
46 
2116 
7 
49 
17 
289 
27 
729 
37 
1369 
47 
2209 
8 
64 
18 
324 
28 
784 
38 
1444 
48 
2304 
9 
81 
19 
361 
29 
841 
39 
1521 
49 
2401 
10 
100 
20 
400 
30 
900 
40 
1600 
50 
2500 
What are the Properties of Square Numbers?
As we have already seen the list of squares from 1 to 50 from the above section, the following properties of square numbers can be generalised:
 Square numbers end with 0, 1, 4, 5, 6 or 9 at the unit’s place
 If a number ends with 1 or 9, then its square will always end with 1
 If a number ends with 4 or 6, then its square will always end with 6
 Unit digit of square of any number will be the unit digit of square of its last digit
 The square root of perfect square is always a natural number
 The perfect squares will end with even numbers of zeros
 Square of even numbers are always even
 Square of odd numbers are always odd
Let us discuss all these properties of square numbers with examples.
Square numbers end with 0, 1, 4, 5, 6 or 9
If we check the squares of numbers from 1 to 10, the unit digit of the square numbers will have 0, 1, 4, 5, 6 or 9. Thus, for all the perfect squares, the unit digit will consist of only 0, 1, 4, 5, 6 or 9 and none of the square numbers will end with 2, 3, 7 or 8.
Examples:

Square of Numbers ending with 1 or 9 always end with 1
According to this property, if the number ends with 1 or 9, then the square of the number always ends with 1 at the unit place. See some examples below:
Examples:

Square of Numbers ending with 4 or 6 always end with 6
As per this property of square numbers, if a number ends with 4 or 6 at the unit’s place, then the square of that number will always end with 6 at the unit place. The examples are:

Unit digit of square of number is the same as unit digit of square of its last digit
This property explains the square of any number such as a twodigit number will have the same digit at unit place, as the square of its unit digit will have.
For example, the square of 23 is 529
23^{2} = 529
The unit place of 23 has 3 and unit place of 529 has 9
Square of 3 is equal to 9
Square root of perfect square is always a natural number
Examples of perfect squares are 4, 9, 16, 25, etc.
Hence, if we take the square root of these perfect squares, we will get a natural number only.

Therefore, square root is the inverse method of finding the square of a number.
Perfect squares will end with even numbers of zeros
If a number ends with zero at unit place, then the square of such a number will always end with even numbers of zeros. See the example below to understand this property.
Examples:

Square of even numbers are always even
As per the property, the square of even numbers will always result in an even number. Thus,
(2n)^{2} = 4n^{2}
Where n is any natural number.
Examples:

Square of odd numbers are always odd
If we find the square of any odd number, the result will always be an odd number. Thus,
(2n + 1)^{2} = 4(n^{2} + n) + 1
Where n is any natural number
Examples:

Squares and Square Root Related Articles
 Square and Square Root
 Square Root 1 to 100
 Square Root Of A Number By Repeated Subtraction
 Square Root And Cube Root
 Square Root Questions
 Sum of Squares
 How to Find Square Root
Frequently Asked Questions on Properties of Square Numbers
What are the four properties of square numbers?
Square numbers always end with digits 0, 1, 4, 5, 6 or 9, at its unit place. Square of a number ending with 4 and 6, will always end with 6 at unit place. If a number has 1 or 9 in the unit’s place, then it’s square ends in 1. Square numbers can only have an even number of zeros at the end.
What is the square of number 35?
The square number of 35 is 1225.
What is the property of square of even numbers?
The square of even numbers are always even numbers.
What is the property of a square of odd numbers?
The square of odd numbers are always odd numbers.
Is 49 a perfect square or not?
49 is a perfect square because the root of 49 is equal to 7.