Is 1306 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 1306 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 1306 has more than two factors, it is not a prime number. Several methods are used to distinguish between prime and composite numbers. A few methods are:

• 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 1306 is prime or composite.

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

Step 2: Divide 1306 by 2. It is divisible by 2, so 2 is a factor of 1306.

Step 3: Divide 1306 by 3. It is not divisible by 3, so 3 is not a factor of 1306.

Step 4: You can simplify checking divisors up to the root value of 1306. We then need to only check divisors up to the root value.

Step 5: When we divide 1306 by 2, 653, and others, it is divisible by 2 and 653. Since 1306 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 6. Six is an even number, which means that 1306 is divisible by 2.

Divisibility by 3: The sum of the digits in the number 1306 is 10. Since 10 is not divisible by 3, 1306 is also not divisible by 3.

Divisibility by 5: The unit’s place digit is 6. Therefore, 1306 is not divisible by 5.

Divisibility by 7: Use the divisibility rule for 7 to check further.

Divisibility by 11: In 1306, the sum of the digits in odd positions is 1 + 0 = 1, and the sum of the digits in even positions is 3 + 6 = 9. The difference is 8, which means that 1306 is not divisible by 11. Since 1306 is divisible by more than just 1 and 1306 itself, 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 the following steps.

Step 1: Write numbers in a grid.

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 have a list of prime numbers. Since 1306 is not present in the list of prime numbers, 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 1306 as 2 × 653. Step 2: In 2 × 653, 653 is a prime number. Therefore, no further factorization is possible.

Step 3: Now we get the product consisting of prime numbers. Hence, the prime factorization of 1306 is 2 × 653.

• Composite numbers: Natural numbers greater than 1 that are divisible by more than 2 numbers are called composite numbers. For example, 12 is a composite number because 12 is divisible by 1, 2, 3, 4, 6, and 12.
   
• Prime numbers: Natural numbers greater than 1 that have no divisors other than 1 and themselves. For example, 5 is a prime number because its only divisors are 1 and 5.
   
• Sieve of Eratosthenes: An ancient algorithm used to find all prime numbers up to a specified integer.
   
• Factors: The numbers that divide the number exactly without leaving a remainder are called factors. For example, the factors of 4 are 1, 2, and 4 because they divide 4 completely.
   
• Divisibility rules: A set of rules that help determine if one number is divisible by another without performing full division.