# 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 2×2 + 2x + 1 is a perfect square or not part of list of perfect squares?:

• 4×2+2x+1
• 4×2 + 2x + (14 + 34)
• {4×2+2x + 14} + 34
• {4×2+2x + (12 x 12) + 34
• {4×2+2x+( 12 )2} + 34
• {(2x)2+2(2x) x 12 + (12)2 + 34
• (2x + 12)2+ 34
• Therefore, 4×2 + 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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