Difference between regular TV and Smart TV
Smart TVs have really taken the world by storm. With so many great features and the ability to connect to the Internet, technology has changed the way we watch TV.
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
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).
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.
A perfect square number divisible by a prime number p will also be divisible by p^2, and vice versa.
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.
For example:
For example: The square number 18 is divisible by 3, then it will also be divisible by the square of 3, which is 9.
There are 2 types of square numbers:
Even square number | Odd square number |
A perfect square number is even if and only if it is the square of an even integer. | A perfect square number is odd if and only if it is the square of an odd integer. |
For example, the number 36 is an even square number because it is the square of the number 6 (an even number). | For example, the number 25 is an even square number because it is the square of the number 5 (an odd 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 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:
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 ...
Smart TVs have really taken the world by storm. With so many great features and the ability to connect to the Internet, technology has changed the way we watch TV.
Refrigerators are familiar appliances in families. Refrigerators usually have 2 compartments, the cool compartment is spacious and has a light that automatically turns on every time the user opens it, while the freezer compartment is narrow and has no light.
Wi-Fi networks are affected by many factors beyond routers, bandwidth, and interference, but there are some smart ways to boost your network.
If you want to go back to stable iOS 16 on your phone, here is the basic guide to uninstall iOS 17 and downgrade from iOS 17 to 16.
Yogurt is a great food. Is it good to eat yogurt every day? What will happen to your body when you eat yogurt every day? Let's find out together!
This article discusses the most nutritious types of rice and how to maximize the health benefits of whichever rice you choose.
Establishing a sleep schedule and bedtime routine, changing your alarm clock, and adjusting your diet are some of the measures that can help you sleep better and wake up on time in the morning.
Rent Please! Landlord Sim is a simulation mobile game on iOS and Android. You will play as a landlord of an apartment complex and start renting out an apartment with the goal of upgrading the interior of your apartments and getting them ready for rent.
Get Bathroom Tower Defense Roblox game codes and redeem them for exciting rewards. They will help you upgrade or unlock towers with higher damage.
Let's learn about the structure, symbols and operating principles of transformers in the most accurate way.
From better picture and sound quality to voice control and more, these AI-powered features are making smart TVs so much better!
DeepSeek initially had high hopes. As an AI chatbot marketed as a strong competitor to ChatGPT, it promised intelligent conversational capabilities and experiences.
It's easy to miss important details when you're jotting down other essentials, and trying to take notes while chatting can be distracting. Fireflies.ai is the solution.
Axolot Minecraft will be a great assistant for players when operating underwater if they know how to use them.
A Quiet Place: The Road Ahead's configuration is rated quite highly, so you will need to consider the configuration before deciding to download.