What is a perfect square number? How to identify it and detailed examples

A perfect square number is a number that is equal to the square of an integer. Let's learn about the properties, how to recognize and calculate perfect square numbers in the article below to better understand this type of number.

Table of Contents

• What is a perfect square number?
  • How to identify square numbers
  • Divisibility of square numbers
  • Smallest square number
  • Largest square number
  • Identical constant to calculate the difference of two square numbers
  • Characteristics of square numbers
• Square numbers
• Examples of perfect square numbers
• Exercises on square numbers

What is a perfect square number?

A perfect square number is a number that is the exact square of an integer. Or simply put, a perfect square number is a natural number whose square root is also a natural number.

Integers include positive integers (1, 2, 3,…), negative integers (-1, -2, -3,…) and 0. The set of integers is denoted by Z.

However, the square root of a square number has only natural values, that is, positive integers.

For example:

The number 4 is a perfect square because the square of the number 2 is 4.

9 is a perfect square number, (because 9 is equal to the square of 3).

How to identify square numbers

1. Look at the last digit : The last digit of a perfect square number is 0, 1, 4, 5, 6, 9. Numbers that end in 2, 3, 7, 8 are not called perfect square numbers.

2. Look at the last digit: A perfect square number can only have 1 of 2 forms: 4n or 4n + 1, no perfect square number has the form 4n + 2 or 4n + 3 (with n € N).

For example: Suppose n = 1 then the square number is in the form 4 x n = 4. Or n = 2 then the square number is in the form 4 x 2 + 1 = 9.

It cannot be in the form 4 x 2 + 2 = 10 or 4 x 2 + 3 = 11.

3. The tens digit of a perfect square number is even if the last digit is 1 or 9.

For example: The square number 81 (square of 9).

4. The tens digit of a perfect square number ending in 5 is 2.

For example: The square number 225 (square of 15).

5. If the perfect square number ends in 4, the tens digit is an even number.

For example: The square number 64 (square of 8).

6. If the square number ends in 6, the tens digit is odd.

For example: The square number 16 (square of 4).

7. When factored into prime numbers, a perfect square number contains only prime factors with even exponents.

For example: The square number 16 = 2 x 2 x 2 x 2 = 2 ^ 4.

Divisibility of square numbers

A perfect square number divisible by a prime number p will also be divisible by p^2, and vice versa.

• A perfect square number divisible by 2 is divisible by 2^2 = 4.
• A perfect square number divisible by 3 is divisible by 3^2 = 9.
• A perfect square number divisible by 5 is divisible by 5^2 = 25.
• A perfect square number divisible by 8 (= 2^3) is also divisible by 2^4 = 16 (written as a power of a number).
• The square number 36 ( 6^2 ) is divisible by 2 => 36 is divisible by 4 ( 2^2 )
• The square number 144 (12^2) is divisible by 3 (144:3=48) => 144 is divisible by 9 (144:9=16)

Smallest square number

The smallest perfect square number in the set of perfect square numbers is 0. In the number range from 0 to 100, there are 10 perfect square numbers less than 100. They include the numbers: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81.

Largest square number

• The largest 1-digit square number is 9.
• The largest two-digit square number is 81.
• The largest 3-digit square number is 312.
• The largest 4-digit square number is 9801
• The largest 5-digit square number is 99856

Identical constant to calculate the difference of two square numbers

For example:

Characteristics of square numbers

• Formula to calculate the difference of two square numbers: a^2 - b^2 = (ab)(a+b).
• If a square number is divisible by a prime number, then it will also be divisible by the square of that prime number.

For example: The square number 18 is divisible by 3, then it will also be divisible by the square of 3, which is 9.

Square numbers

There are 2 types of square numbers:

Examples of perfect square numbers

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 square numbers.

Exercises on square numbers

Lesson 1 : In the following series of numbers, which is a perfect square number: 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.

Lesson 3 : Prove that the number B = 4n^4 + 4n³ + n² is a perfect square for every positive integer n.

Solution:

B = 4n^4 + 4n³ + n²= n²(4n² + 4n + 1)= n²(2n + 1)²

We see that B can be expressed as a product of two squares. Or B = [n(2n+1)]², and n(2n + 1) is an integer. So we conclude that B is a perfect square.

Lesson 4: 

Find a natural number n such that the following number is a perfect square: B = n² + 4n + 1.

Solution:

Since the number B is a perfect square, we set n² + 4n + 1 = b²

= 4n²+16n+4=4b²

= (4n²+16n+16)-16+4=4b²

= (2n+4)²- 4b² = 12

= (2n+4+2b)x(2n+4-2b)=12

Notice that 2n+4+2b 2n+4-2b, and these are all positive integers. So we can find the corresponding pairs of numbers: (12, 1), (6, 2) and (4, 3). You need to consider each case to find n and b. Specifically:

• Case 1: (2n + 4 + 2b) (2n + 4 - 2b) = 12 = 12 x 1 = n = 5/4, b = 11/4
• Case 2: (2n + 4 + 2b) (2n + 4 - 2b) = 12 = 6 x 2 = n = 0, b = 1
• Case 3: (2n + 4 + 2b) (2n + 4 - 2b) = 12 = 4 x 3 = n = -1/4, b = 1/4

But n is a natural number, so only the answers n = 0, b = 1 are satisfactory. And n = 0, so the square number B = 1.

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 issues about perfect square numbers.

In addition to square numbers, you can learn about other types of numbers in mathematics such as mixed numbers , fractions ...