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List of perfect squares - introduction and examples

What are perfect squares? a rational number that is equal to the square of another rational number. A rational number is a number that can be expressed exactly by a ratio of two integers. Below is an example for list of perfect squares:

  • 3 × 3 =9. Thus: 9 is among the list of perfect squares
  • 2 × 2 = 4. Thus: 4 is also among the list of perfect squares

List of perfect squares means those numbers made by squaring a whole number. If the product of 2 equal integers successfully gives a square number, then the 2 equal or similar whole numbers are perfect squares. For instance, let’s say we have 25 which is a perfect square that would be obtained from the product or multiplication of two equal integers or whole numbers such as 5, we can call this whole number a perfect square

A little trick that can be used to make the best list of perfect squares. You can easily take the square root of any whole number and in the end, we will still have the whole number or the previously squared integers which gave us the whole number that we used to find its square root. Using the example above, the square root of 25 will give 5. Remember that 5 was multiplied by itself to obtain the 25. i.e. √25 = 5 also, 52 will give 25

Furthermore, it is important to remember that an integer or whole number cannot have a fractional part which means that it is impossible to find any fractional number among the list of perfect squares

 

List of perfect squares and their reasons for being perfect squares

The product of any whole number by itself is given a unique name due to its geometrical analysis. Let’s imagine a rectangle and its dimensions. You notice that it’s not a complete square or let’s say it’s unlike the dimensions that can be found in a square. This means that all rectangular shaped objects are imperfect rectangle or we could even further say that they are not or will never be a perfect square and the only reason behind this is due to the difference in their dimension. Now, imagine this rectangle has the same dimensions as well as its length and height, then we can call it a perfect square

Below is a diagrammatical representation with a little explanation of the above description of list of perfect squares:

Take a quick peek at some of the list of perfect squares ranging from 1 till 100 below:

  •  

    Integers

     

    Square of the integers (i.e. integer multiplied by itself)

     

    List of perfect squares

     

    1

     

    1 × 1

     

    1

     

    2

     

    2 × 2

     

    4

     

    3

     

    3 × 3

     

    9

     

    4

     

    4 × 4

     

    16

     

    5

     

    5 × 5

     

    25

     

    6

     

    6 × 6

     

    36

     

    7

     

    7 × 7

     

    49

     

    8

     

    8 × 8

     

    64

     

    9

     

    9 × 9

     

    81

     

    10

     

    10 × 10

     

    100

     

    11

     

    11 × 11

     

    121

     

    12

     

    12 × 12

     

    144

     

    13

     

    13 × 13

     

    169

     

    14

     

    14 × 14

     

    196

     

    15

     

    15 × 15

     

    225

     

    16

     

    16 × 16

     

    256

     

    17

     

    17 × 17

     

    289

     

    18

     

    18 × 18

     

    324

     

    19

     

    19 × 19

     

    361

     

    20

     

    20 × 20

     

    400

     

    21

     

    21 × 21

     

    441

     

    22

     

    22 × 22

     

    484

     

    23

     

    23 × 23

     

    529

     

    24

     

    24 × 24

     

    576

     

    25

     

    25 × 25

     

    625

     

    26

     

    26 × 26

     

    676

     

    27

     

    27 × 27

     

    729

     

    28

     

    28 × 28

     

    784

     

    29

     

    29 × 29

     

    841

     

    30

     

    30 × 30

     

    900

     

    31

     

    31 × 32

     

    961

     

    32

     

    32 × 32

     

    1024

     

    33

     

    33 × 33

     

    1089

     

    34

     

    34 × 34

     

    1156

     

    35

     

    35 × 35

     

    1225

     

    36

     

    36 × 36

     

    1296

     

    37

     

    37 × 37

     

    1369

     

    38

     

    38 × 38

     

    1444

     

    39

     

    39 × 39

     

    1521

     

    40

     

    40 × 40

     

    1600

     

    41

     

    41 × 41

     

    1681

     

    42

     

    42 × 42

     

    1764

     

    43

     

    43 × 43

     

    1849

     

    44

     

    44 × 44

     

    1936

     

    45

     

    45 × 45

     

    2025

     

    46

     

    46 × 46

     

    2116

     

    47

     

    47 × 47

     

    2209

     

    48

     

    48 × 48

     

    2304

     

    49

     

    49 × 49

     

    2401

     

    50

     

    50 × 50

     

    2500

     

    51

     

    51 × 51

     

    2601

     

    52

     

    52 × 52

     

    2704

     

    53

     

    53 × 53

     

    2809

     

    54

     

    54 × 54

     

    2916

     

    55

     

    55 × 55

     

    3025

     

    56

     

    56 × 56

     

    3136

     

    57

     

    57 × 57

     

    3249

     

    58

     

    58 × 58

     

    3364

     

    59

     

    59 × 59

     

    3481

     

    60

     

    60 × 60

     

    3600

     

    61

     

    61 × 61

     

    3721

     

    62

     

    62 × 62

     

    3844

     

    63

     

    63 × 63

     

    3969

     

    64

     

    64 × 64

     

    4096

     

    65

     

    65 × 65

     

    4225

     

    66

     

    66 × 66

     

    4356

     

    67

     

    67 × 67

     

    4489

     

    68

     

    68 × 68

     

    4624

     

    69

     

    69 × 69

     

    4761

     

    70

     

    70 × 70

     

    4900

     

    71

     

    71 × 71

     

    5041

     

    72

     

    72 × 72

     

    5184

     

    73

     

    73 × 73

     

    5329

     

    74

     

    74 × 74

     

    5476

     

    75

     

    75 × 75

     

    5625

     

    76

     

    76 × 76

     

    5776

     

    77

     

    77 × 77

     

    5929

     

    78

     

    78 × 78

     

    6084

     

    79

     

    79 × 79

     

    6241

     

    80

     

    80 × 80

     

    6400

     

    81

     

    81 × 81

     

    6561

     

    82

     

    82 × 82

     

    6724

     

    83

     

    83 × 83

     

    6889

     

    84

     

    84 × 84

     

    7056

     

    85

     

    85 × 85

     

    7225

     

    86

     

    86 × 86

     

    7396

     

    87

     

    87 × 87

     

    7569

     

    88

     

    88 × 88

     

    7744

     

    89

     

    89 × 89

     

    7921

     

    90

     

    90 × 90

     

    8100

     

    91

     

    91 × 91

     

    8281

     

    92

     

    92 × 92

     

    8464

     

    93

     

    93 × 93

     

    8649

     

    94

     

    94 × 94

     

    8836

     

    95

     

    95 × 95

     

    9025

     

    96

     

    96 × 96

     

    9216

     

    97

     

    97 × 97

     

    9409

     

    98

     

    98 × 98

     

    9604

     

    99

     

    99 × 99

     

    9801

     

    100

     

    100 × 100

     

    10000

     

Stepwise explanation with list of perfect squares solved examples

Step 1: a perfect square never ends in 2, 3, 7 or 8. This is the first thing to consider while checking integers or whole numbers if they are actually perfect squares or not

Step 2: calculate and get the digital root of the given number. This digital root helps check the number if it is among the list of perfect squares. A perfect square will surely have a digital root of: 0, 1, 4 or 7. E.g. 15626 for instance ends with 6 which satisfy our rule number one which means that the number is a perfect square or part of list of perfect squares?

Example 1: Can this equation x2 + 10x + 25 be a perfect square or part of list of perfect squares?:

  • x2 + 10x + 25
  • x2 + 10x + (5 × 5)
  • x2 + 10x + 52
  • x2 + 2(5 × x) + 52
  • (x + 5)2
  • Therefore, x2 + 10x + 25 is a perfect square or part of list of perfect squares?

Example 2: do you think this equation 2x2 + 2x + 1 is a perfect square or not part of list of perfect squares?:

  • 4x2+2x+1
  • 4x2 + 2x + (14 + 34)
  • {4x2+2x + 14} + 34
  • {4x2+2x + (12 x 12) + 34
  • {4x2+2x+( 12 )2} + 34
  • {(2x)2+2(2x) x 12 + (12)2 + 34
  • (2x + 12)2+ 34
  • Therefore, 4x2 + 2x + 1 is not a perfect square or part of list of perfect squares?

How to check if a number is among the list of perfect squares with solved examples

 

There are few properties that can be used to check if a number or a whole number is a perfect square or not. The ways to check whether or not a number is a perfect square include the following. All perfect squares end in either of the following integers or whole numbers: 1, 4, 5, 6, 9 or 0. Therefore, any number which ends in either of the following: 2, 3, 7 or 8 are not perfect squares. Furthermore, for all the numbers ending in 1, 4, 5, 6, & 9 and for numbers ending in even (0s) zeros as well, we remove the (0s) zeros that ends the number or integer then use the following tests:

  • No number can be referred as been a perfect square except its digital root is among the following numbers: 1, 4, 7, or 9. To determine the digital root of any number, sum up all of its digits and if your answer is more than 9, then add the digits of the answer that you got. Lastly, your final and the single digit you get or will get at of the summing up will be your digital root of the number
  • If a unit digit ends with number 5, ten’s digit will always be 2
  • Again, if the unit digit ends in 6, ten’s digit will also and always be odd numbers such as: 1, 3, 5, 7, and 9 otherwise, it will always be even in cases where the unit digit ends in: 1, 4, and 9 then our ten’s digit is always even. i.e. 2, 4, 6, 8, 0
  • If a number is divisible by 4, its square will have no other remainder other than (0) zero when divided by 8
  • Squaring even numbers which is cannot be divided by 4 gives a remainder of 4 while the square of any odd number will always give a remainder of 1 when divided by 8
  • Total numbers of prime factors of a perfect square are always odd

Example 1: can this number - 4539 - be a perfect square or even be among the list of perfect squares?:

  • the number 4539 ends with a 9. Remember from our rules above, let’s find its digital root, i.e. 4 + 5+ 3 + 9 = 21. The answer is greater than 9
  • Next is to add or sum up the answer digits, i.e. 2 + 1 = 3. Now we have an answer that is less than 9. From our rules, digit sum is 3 this means that 4539 is not a perfect square

Example 2: do you think 5776 can be among the list of perfect squares?:

  • the number 5776 ends with a 6. Remember from our rules above, let’s find its digital root, i.e. 5 + 7+ 7 + 6 = 25. The answer is greater than 9
  • Next is to add or sum up the answer digits, i.e. 2 + 5 = 7. Now we have an answer that is less than 9. From our rules, digit sum is 7 this means that 5776 is or may be considered as a perfect square or among the list of perfect squares
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