Is 1307 a Prime Number?

There are two types of numbers, mostly

• prime numbers
   
• composite numbers

depending on the number of factors. A prime number is a natural number that is divisible only by 1 and itself. For example, 3 is a prime number because it is divisible by 1 and itself. A composite number is a positive number that is divisible by more than two numbers. For example, 6 is divisible by 1, 2, 3, and 6, making it a composite number.

Prime numbers follow a few properties like: Prime numbers are positive numbers always greater than 1. 2 is the only even prime number. They have only two factors: 1 and the number itself. Any two distinct prime numbers are co-prime numbers because they have only one common factor, which is 1. As 1307 has more than two factors, it is not a prime number.

The characteristic of a prime number is that it has only two divisors: 1 and itself. Since 1307 has more than two factors, it is not a prime number. A few methods are used to distinguish between prime and composite numbers. These methods include:

• Counting Divisors Method
   
• Divisibility Test
   
• Prime Number Chart
   
• Prime Factorization

The method in which we count the number of divisors to categorize the numbers as prime or composite is called the counting divisors method. Based on the count of the divisors, we categorize prime and composite numbers. If there is a total count of only 2 divisors, then the number would be prime. If the count is more than 2, then the number is composite. Let’s check whether 1307 is prime or composite.

Step 1: All numbers are divisible by 1 and itself.

Step 2: Divide 1307 by 2. It is not divisible by 2, so 2 is not a factor of 1307.

Step 3: Divide 1307 by 3. It is not divisible by 3, so 3 is not a factor of 1307.

Step 4: You can simplify checking divisors up to 1307 by finding the approximate square root value, which is around 36. We then need to only check divisors up to this value.

Step 5: When we divide 1307 by 7, it is divisible, so 7 is a factor of 1307. Since 1307 has more than 2 divisors, it is a composite number.

We use a set of rules to check whether a number is divisible by another number completely or not. It is called the Divisibility Test Method.

Divisibility by 2: The number in the ones' place value is 7. Seven is an odd number, which means that 1307 is not divisible by 2.

Divisibility by 3: The sum of the digits in the number 1307 is 11. Since 11 is not divisible by 3, 1307 is also not divisible by 3.

Divisibility by 5: The unit’s place digit is 7. Therefore, 1307 is not divisible by 5.

Divisibility by 7: Perform the divisibility check for 7 to find that 1307 is divisible by 7. Since 1307 is divisible by 7, it has more than two factors. Therefore, it is a composite number.

The prime number chart is a tool created by using a method called “The Sieve of Eratosthenes.” In this method, we follow these steps:

Step 1: Write 1 to 100 in 10 rows and 10 columns.

Step 2: Leave 1 without coloring or crossing, as it is neither prime nor composite.

Step 3: Mark 2 because it is a prime number and cross out all the multiples of 2.

Step 4: Mark 3 because it is a prime number and cross out all the multiples of 3.

Step 5: Repeat this process until you reach the table consisting of marked and crossed boxes, except 1. Through this process, we will have a list of prime numbers from 1 to 100. The list includes numbers like 2, 3, 5, 7, and so on. Since 1307 is not on this chart, and further analysis shows it is divisible by 7, it is a composite number.

Prime factorization is a process of breaking down a number into prime factors. Then multiply those factors to obtain the original number.

Step 1: We can write 1307 as 7 × 187.

Step 2: In 7 × 187, 187 is a composite number. Further, break down 187 into 11 × 17.

Step 3: Now we get the product consisting of only prime numbers. Hence, the prime factorization of 1307 is 7 × 11 × 17.

• Composite numbers: Natural numbers greater than 1 that are divisible by more than 2 numbers are called composite numbers. For example, 1307 is a composite number because it is divisible by 1, 7, 11, 17, 77, 119, 187, and 1307.
   
• Divisibility test: A method to determine if one number can be divided by another without leaving a remainder.
   
• Prime factorization: A process of expressing a number as a product of its prime factors.
   
• Factors: The numbers that divide the number exactly without leaving a remainder are called factors. For example, the factors of 1307 are 1, 7, 11, 17, 77, 119, 187, and 1307.
   
• Sieve of Eratosthenes: An ancient algorithm used to find all primes up to a specified integer.