Properties of Square Numbers

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:

22 = 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 N2 is the square of N. The list will help us to learn about the properties of square numbers.

N

N2

N

N2

N

N2

N

N2

N

N2

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:

  • 12 = 1
  • 22 = 4
  • 32 = 9
  • 42 = 16
  • 52 = 25
  • 62 = 36
  • 72 = 49
  • 82 = 64
  • 92 = 81
  • 102 = 100

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:

  • 112 = 121
  • 212 = 441
  • 192 = 361
  • 292 = 841

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:

  • 142 = 196
  • 162 = 256
  • 242 = 576

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 two-digit 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

232 = 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.

  • √4 = 2
  • √9 = 3
  • √16 = 4
  • √25 = 5

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:

  • 102 = 100
  • 202 = 400
  • 502 = 2500
  • 1002 = 10000
  • 7002 = 4900

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 = 4n2

Where n is any natural number.

Examples:

  • 22 = 4
  • 62 = 36
  • 122 = 144
  • 182 = 324
  • 302 = 900

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(n2 + n) + 1

Where n is any natural number

Examples:

  • 72 = 49
  • 152 = 225
  • 232 = 529
  • 452 = 2025

Squares and Square Root Related Articles

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.

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