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? How to recognize and detailed examples Picture 1

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

1. A perfect square number divisible by prime number p is also divisible by p^2, and vice versa.
2. A perfect square number is divisible by 2 then it is divisible by 2^2 = 4.
3. A perfect square number is divisible by 3 then it is divisible by 3^2 = 9.
4. A perfect square number is divisible by 5 then it is divisible by 5^2 = 25.
5. A square number divisible by 8 (= 2^3) is also divisible by 2^4 = 16 (written as a power of a number).
6. Square number 36 ( 6^2 ) is divisible by 2 => 36 is divisible by 4 ( 2^2 )
7. 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

1. The largest 1-digit perfect square number is 9
2. The largest 2-digit square number is 81
3. The largest perfect square number with 3 digits is 312
4. The greatest square number with 4 digits is 9801
5. The largest perfect square number with 5 digits is 99856

Constant equality to calculate the difference of two perfect square numbers

What is a perfect square number? How to recognize and detailed examples Picture 2

For example: What is a perfect square number? How to recognize and detailed examples Picture 3

Characteristics of perfect square numbers

1. Formula to calculate the difference of two perfect square numbers: a^2 - b^2 = (ab)(a+b).
2. 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:

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.

David Pac

Update 11 April 2024