What are prime numbers? Prime number chart in mathematics and examples

Prime numbers, a fundamental concept in elementary and high school math. But what exactly are they? Mytour helps you explore prime numbers, including charts, sequences, and superprime numbers.

What are prime numbers?

Prime numbers form a set of integers greater than 1 that can only be evenly divided by themselves and 1. For example, 3 is only divisible by 1 and itself. Similarly, numbers like 11, 13, 17, 19, and so on, are prime numbers.

Basic Properties of Prime Numbers

Characteristics of prime numbers: What are the key features of prime numbers?

• Does not include the number 1.
• All prime numbers, except for 2, are not divisible by 2.
• The sequence of prime numbers is infinite and has no limits, similar to the set of natural numbers N.
• The smallest divisor of any number in the set N is always a prime number. For example, 15 is divisible by 3 (the smallest divisor of 15), and 3 is a prime number.
• The smallest divisor of a positive number X, which is different from 1 and less than the square root of X, is a prime number. For example, 30 has the smallest divisor of 5, and since 5 is different from 1 and less than the square root of 30, it is considered a prime number.

Prime number chart in mathematics

As shared, the sequence of prime numbers in mathematics has no specific limit due to the infinite nature of natural numbers. However, there are common numbers used frequently. The following is the prime number chart containing numbers less than 1000:

Illustrative example exercises on prime numbers

To help you grasp the concept of prime numbers and the prime number sequence better, refer to the following illustrative exercises. These include two types of exercises: prime factorization and finding the greatest common divisor. The specific types of exercises are as follows:

Prime Factorization Exercises

Prime factorization is the process of identifying the smallest prime number that cannot be divided evenly. In this case, you examine if the given number A is divisible by 2, then 3, 5, and so on, from the prime number sequence. Finally, you multiply the identified prime factors together.

For example, if the task is to factorize the number 140 into prime factors, you divide 140 by the smallest prime number, 2, resulting in 70. Continuing to divide 70 by 2 gives 35. At this point, 35 cannot be divided by 2 anymore but can be divided by 5, resulting in 7, a prime number divisible only by 1 and itself. So, the prime factors at this point are 2, 2, 5, 7, and 1, written as: 140 = 2 x 2 x 5 x 7 x 1.

Finding the Greatest Common Divisor Exercises

Finding the greatest common divisor (GCD) often appears in problems related to prime numbers. In such cases, a number given in the problem may have multiple common divisors, denoted as GCD(a, b), where a and b are the numbers to find the GCD for. To solve these problems, factorize the numbers into prime factors and take the minimum exponent for each prime factor.

For example, tasked with finding the greatest common divisor of 120 and 80, analyze them similarly as discussed earlier. The correct result is 80 = 2^4 x 5 and 120 = 2^3 x 3 x 5.

Looking at the results, you notice that the common factors of 120 and 80 are 2 and 5. Proceed to analyze the exponents, revealing that 2 has a minimum exponent of 3, and 5 has a minimum exponent of 1. Therefore, the greatest common divisor is determined as GCD(80, 120) = 2^3 x 5 = 40.

Another Definition

Alongside understanding the basic concepts and properties answering what prime numbers are, delve into additional definitions related to prime numbers, coprime numbers, superprime numbers, and the question of whether 1 and 2 are prime numbers. Here are definitions related to prime numbers:

Are 1 and 2 Prime Numbers?

By definition, a number greater than 1, not divisible by any number other than itself, is called a prime number. Therefore, among the numbers 1 and 2, only the number 2 satisfies the conditions to be called a prime number.

What Does Coprime Mean?

Numbers are considered coprime if the largest number that both divide evenly is 1. For example, 2 and 3 are coprime because the largest number they both divide evenly is 1. On the other hand, the numbers 18 and 54 are not considered coprime because the largest number they can both divide evenly is 9.

What Are Superprime Numbers?

Superprime numbers are a type of number where adding any digit next to them still results in a prime number. For instance, 73 is a prime number, and adding the digit 3 next to it results in 733, which is still a prime number. Similarly, adding any digit, such as 7, results in 7337, which is also a prime number.

Here are the fundamental concepts of what prime numbers are, some common related math problems, and the tables, sequences of prime numbers, and superprime numbers. We hope that through this knowledge, you can conveniently apply it to your learning process. If you want to learn more about mathematics, continue to the next articles on Mytour.

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