What is a prime number? Definition, Types, Sample Problems

The method used to represent and work with numbers is known as the number system. A number system is a system of writing to represent numbers. It is the mathematical notation used to represent numbers of a given set by using digits or other symbols. It allows operating arithmetic operations such as division, multiplication, addition, subtraction. Some important number systems are Decimal Number System, Binary Number System, Octal Number System, Hexadecimal Number System.

Decimal Number System

The decimal number system consists of ten digits i.e. from 0 to 9. The base of the decimal number system is 10. These digits can be used to represent or express any numeric value. For example, the decimal number 153 consists of the digit 3 in one place, the digit 5 in the tens place, and the digit 1 in hundreds place which can be represented as,

(1×10² ) + (5 × 10¹) + (3  × 10⁰)

= (1 × 100) + (5 × 10) + (3 × 1)

 {where, 100 = 1}

= 100 + 50 + 3

= 153

There are different types in decimal number systems based on the different characteristics, for instance, there are whole numbers, natural numbers, prime numbers, composite numbers, etc. Let’s learn about Prime numbers in detail,

What is a Prime number?

In the number system, Prime Numbers are those numbers that have only two factors that is 1 and the number itself. In other words, a prime number is that number that is exactly divisible by 1 and the number itself. 

• A Prime number should contain exactly two factors.
• A prime number should be divisible 1 and the number itself.

Let’s assume p is a prime number then p has only 2 factors that are 1 and p itself. Any number which does not follow this is termed a composite number. For example factors of 8 are 1, 2, 4, and 8, which are four factors in total. But factors of 5 are 1 and 5 itself, totally two factors. Hence, 5 is a prime number but 8 is not a prime no, instead, it is a composite number.

First Ten Natural Prime Numbers are – 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

1. Factors of 1 are =1  ( Not Prime Number because it has only one factor)
2. Factors of 2 are = 1 and 2 ( Prime Number because it has only two factors )
3. Factors of 3 are =1 and 3 ( Prime Number because it has only two factors )
4. Factors of 4 are =1, 2, and 4 ( Not Prime Number because it has three factors )
5. Factors of 5 are =1 and 5 ( Prime Number because it has only two factors )
6. Factors of 6 are =1, 2, 3, and 6 ( Not Prime Number because it has four factors )
7. Factors of 7 are =1 and 7 ( Prime Number because it has only two factors )
8. Factors of 8 are =1, 2, 4, and 8 ( Not Prime Number because it has four factors )
9. Factors of 9 are =1, 3, and 9 ( Not Prime Number because it has three factors )
10. Factors of 10 are =1, 2, 5, and 10 ( Not Prime Number because it has four factors )

NOTE: 1 is a non-prime number because according to the definition, a prime number should contains only two factors but 1 has only one factor. Therefore 1 is not a prime number.

List of Prime Number between 1 to 100

It is known, the prime numbers are the numbers that have only two factors which are 1 and the number itself. The above are the prime numbers that are present between 1 and 100.

Even Prime Numbers

Even prime numbers are the numbers that are evenly divisible by 2. Therefore, 2 is the only prime number in the number system that is even so 2 is known as an even prime number and the remaining prime numbers are the odd numbers, therefore they are called odd prime numbers. 

• Even Prime Numbers = 2
• Odd Prime Numbers = 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 etc.

Twin Prime Numbers

Twin prime numbers are the numbers that have only one composite number between them are known as twin prime numbers or twin primes. The other words of twin prime numbers are the pair of prime numbers whose difference between two prime numbers is 2 only. Few examples are,

• (5, 7) = 7 – 5 = 2
• (11, 13) = 13 – 11 = 2
• (17, 19) = 19 – 17 = 2
• (41, 43) = 43 – 41 = 2

Sample Problems

Question 1: Is 51 a Prime Number ? 

Answer :

No, because the only factors 51 are 1, 3, 17, and 51. According to the definition, a prime number should contain only two factors. Factors of 51 = 1, 3, 17, 51 ( 4 Factors ). So, 51 is not a prime number.

Question 2: Is 1 a Prime Number? 

Answer : 

1 is not a prime number because according to the definition, a prime number should exactly two factors But, number 1 has one and only one factor which is 1 itself. Thus, 1 is not considered a Prime number. Factors of 1 = 1 ( 1 Factor). So, 1 is not a prime number 

Question 3: Is 11 is a Prime Number?

Answer:

Yes, 11 is a prime number because it is only divisible by 2 numbers that is 1 and the number itself (11). Its has two factors 1 and 11 only. Factors of 11 = 1, 11 ( 2 Factors ). So, 11 is a prime number.

Question 4: Find all the prime numbers form the following numbers 1, 22, 3, 51, 75, 88, 65, 63, 19, 7, 39, 47, 60, 100, 12, 10, 5 ?

Answer:

All the prime number from the given numbers are – 3, 19, 7, 39, 47, 5

Explanation,

• Factor of 1 are = 1
• Factor of 22 are = 1, 2, 11, 22
• Factor of 3 are = 1, 3  ( only two factors )
• Factor of 51 are = 1, 3, 17, 51
• Factor of 75 are = 1, 3, 5, 15, 25, 75.
• Factor of 88 are = 1, 2, 4, 8, 11, 22, 44, 88
• Factor of 65 are = 1, 5, 65
• Factor of 63 are = 1, 3, 7, 9 , 21, 63
• Factor of 19 are = 1, 19  ( only two factors )
• Factor of 7 are = 1, 7  ( only two factors )
• Factor of 39 are = 1, 39  ( only two factors )
• Factor of 47 are = 1, 47  ( only two factors )
• Factor of 60 are = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
• Factor of 100 are = 1, 2, 4, 5,10, 20, 25, 50, 100
• Factor of 12 are = 1, 2, 3, 4, 6, 12
• Factor of 10 are = 1, 2, 5, 10
• Factor of 5 are = 1, 5  ( only two factors )

Question 5: Which is the smallest Prime number in the number system?

Answer:

2 is the smallest prime number in the number system. It has only two factors that is 1 and 2. Factors of 2 are : 1, 2

Question 6: What is the largest 2 digit prime numbers?

Answers: 

97 is the largest two digit prime numbers. Factors of 97 are = 1, 97