What is a perfect square number? How to recognize and detailed examples
A perfect square number is the square of a certain natural number. To learn more about the properties, recognition and calculations of perfect square numbers, we invite you to learn in the article below.
What is a perfect square number?
A perfect square number is a number that is equal to the exact square of an integer. Or to put it simply, a perfect square number is a natural number whose square root is also a natural number.
Or to put it simply, a perfect square number is a natural number whose square root is also a natural number.
For example:
The number 4 is a perfect square because the square of the number 2 is 4.
The number 0 is a perfect square number because it is the square of an integer (0=0^2; 0 is an integer) and 0 is the smallest square number.
Properties of perfect square numbers
1. The endings of a perfect square number are 0, 1, 4, 5, 6, 9. In the case of numbers ending in 2, 3, 7, 8, they are not called perfect square numbers.
2. A perfect square number can only have one of two forms: 4n or 4n + 1, there is no perfect square number of the form 4n + 2 or 4n + 3 (with n € N).
For example: Suppose n = 1, then the perfect square number is in the form 4 xn = 4. Or n = 2, then the perfect square number is in the form 4 x 2 + 1 = 9.
Cannot be in the form 4 x 2 + 2 = 10 or 4 x 2 + 3 = 11.
3. A perfect square number can only have one of two forms: 3n or 3n + 1, there is no perfect square number of the form 3n + 2 (with n € N).
4. If a perfect square number ends in 1 or 9, then the tens digit is an even number.
For example: Perfect square number 81 (square of 9).
5. If a perfect square number ends in 5, then the tens digit is 2.
For example: Perfect square number 225 (square of 15).
6. If a perfect square number ends in 4, then the tens digit is an even number.
For example: Perfect square number 64 (square of 8).
7. If a perfect square number ends in 6, then the tens digit is an odd digit.
For example: Perfect square number 16 (square of 4).
8. When factoring prime numbers, square numbers only contain prime factors with even exponents.
For example: Perfect square number 16 = 2 x 2 x 2 x 2 = 2 ^ 4.
Divisibility property of perfect square numbers
- A perfect square number divisible by prime number p is also divisible by p^2, and vice versa.
- A perfect square number is divisible by 2 then it is divisible by 2^2 = 4.
- A perfect square number is divisible by 3 then it is divisible by 3^2 = 9.
- A perfect square number is divisible by 5 then it is divisible by 5^2 = 25.
- A square number divisible by 8 (= 2^3) is also divisible by 2^4 = 16 (written as a power of a number).
- Square number 36 ( 6^2 ) is divisible by 2 => 36 is divisible by 4 ( 2^2 )
- Square number 144 ( 12^2 ) is divisible by 3 (144:3=48) => 144 is divisible by 9 (144:9=16)
The smallest square number
The smallest perfect square number in the set of perfect square numbers is 0. In the series of numbers from 0 to 100, there are all 10 perfect square numbers less than 100. It includes the numbers: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81.
The largest square number
- The largest 1-digit perfect square number is 9
- The largest 2-digit square number is 81
- The largest perfect square number with 3 digits is 312
- The greatest square number with 4 digits is 9801
- The largest perfect square number with 5 digits is 99856
Constant equality to calculate the difference of two perfect square numbers
For example:
Characteristics of perfect square numbers
- Formula to calculate the difference of two perfect square numbers: a^2 - b^2 = (ab)(a+b).
- If a square number is divisible by a prime number, it is also divisible by the square of that prime number.
For example: The perfect square number 18 is divisible by 3 and is also divisible by the square of 3 which is 9.
Forms of perfect square numbers
There are 2 types of perfect square numbers:
Even square numbers | Odd square numbers |
A square number is even if and only if it is the square of an even integer. | A perfect square is odd if and only if it is the square of an odd integer. |
For example: The number 36 is an even perfect square number because it is the square of the number 6 (even number). | For example: The number 25 is an even perfect square number because it is the square of the number 5 (odd number). |
Example of a perfect square number
The numbers 4, 9, 16, 25, 36, 49, 64, 81,100, . are all perfect square numbers.
4 = 2² is an even square number.
9 = 3² is an odd square number.
16 = 4² is an even square number.
25 = 5² is an odd square number.
36 = 6² is an even square number.
49 = 7² is an odd square number.
64 = 8² is an even square number.
81 = 9² is an odd square number.
100 = 10² is an even square number.
Note: Numbers 0 and 1 are also perfect square numbers.
Exercises on perfect square numbers
Lesson 1 : In the following sequence of numbers, what are the perfect square numbers: 9, 81, 790, 408, 121, 380, 2502, 441, 560.
Solution: The perfect square numbers are 9 (3²), 81 (9²), 121 (11²), 441 (21²).
Lesson 2: Prove that the number 1234567890 is not a perfect square number.
Solution: The number 1234567890 is divisible by 5 (because the last digit is 0) but not divisible by 25 (because the last two digits are 90). Therefore the number 1234567890 is not a perfect square number.
Hopefully the above article has provided useful information to help you know what a perfect square number is, whether 0 is a perfect square number as well as the properties and characteristics of a perfect square number. From there, you will have more knowledge to solve problems and problems about square numbers.
In addition to square numbers, you can learn about other types of numbers in mathematics such as mixed numbers, fractions.
You should read it
- The square root formula in Excel - The square root function in Excel
- Quiz: 1 Ha (ha) in how many square meters, square kilometers?
- NASA captures a large, square-shaped piece of ice
- How to download Perfect World VNG for phones and PCs
- Write a program to calculate the square root of a number in Python
- Perfect Cream: Tips for the perfect cake
- SQRT function, how to use square root function in Excel
- How to Enter Square Roots on PC or Mac
May be interested
- Extremely detailed instructions for testing laptop screenshow to recognize if the computer screen is having problems? the article below will help you test the screen in the most detailed and accurate way.
- RANK function - Rank function in Excel - Usage and examplesthe rank function arranges equal numbers of the same rank. however, the duplication will affect the ranking of subsequent numbers. for example, you have the sequence number: 1,2,2,3,4 and are arranged in ascending order, so the order of the number 1 in the series is 5.
- How to use the MAXA function in Excel, detailed examplesthe maxa function in excel helps find the largest value in a data set, including numbers, logical values, and text containing numbers. this is different from the max function, which only counts numbers. let's learn the syntax and usage of the maxa function with practical examples.
- Write a program to calculate the square root of a number in Pythonin this article, tipsmake.com will help you learn more about square roots and how to write a program to calculate square roots in python programming language.
- SQRT function, how to use square root function in Excelthe sqrt function in excel helps us quickly calculate the square root value of a number.
- How to Enter Square Roots on PC or Macthis article shows how to enter the square root sign (√) into text editing programs, including microsoft word, on windows and macos. if you use microsoft word, you can easily insert a square root sign by selecting it from the menu. in other apps, you can use keyboard shortcuts on windows or macos.
- Handle Excel that does not recognize fast - standard number formatswhat is excel not recognizing the number format? how to fix it? instructions for fixing number format errors in excel.
- Perfect Cream: Tips for the perfect cakehere's how to play perfect cream and the note that you need to know in this fun game to spread cream on cake.
- How to calculate the area of a square in Pythonthe tutorial below helps you find the area of a given square in python using different methods.
- How to fix a computer that does not recognize iPhonehow to fix a computer that does not recognize iphone. there are many reasons why your computer does not recognize the iphone, possibly due to the faulty usb cable, the device has not yet received the driver ... let's find out the cause and how to fix it in the article below!