Is 1310 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 1310 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 1310 has more than two factors, it is not a prime number. A few methods are used to distinguish between prime and composite numbers, including:

• 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 numbers as prime or composite.

• 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 1310 is prime or composite.

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

Step 2: Divide 1310 by 2. It is divisible by 2, so 2 is a factor of 1310.

Step 3: Divide 1310 by 3. It is not divisible by 3, so 3 is not a factor of 1310.

Step 4: You can simplify checking divisors up to 1310 by finding the square root value. We only need to check divisors up to this root value.

Step 5: When we divide 1310 by 2, 5, and 10, it is divisible by 2, 5, and 10. Since 1310 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 0. Zero is an even number, which means that 1310 is divisible by 2.

Divisibility by 3: The sum of the digits in the number 1310 is 5. Since 5 is not divisible by 3, 1310 is also not divisible by 3.

Divisibility by 5: The unit’s place digit is 0. Therefore, 1310 is divisible by 5.

Divisibility by 7: The last digit in 1310 is 0. To check divisibility by 7, double the last digit (0 × 2 = 0). Then, subtract it from the rest of the number (131 - 0 = 131). Since 131 is not divisible by 7, 1310 is also not divisible by 7.

Divisibility by 11: In 1310, the sum of the digits in odd positions is 2 (1 + 1), and the sum of the digits in even positions is 4 (3 + 0). The difference is 2, which means that 1310 is not divisible by 11. Since 1310 is divisible by 2 and 5, 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 the following 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 for 1. Through this process, we will have a list of prime numbers from 1 to 100. Since 1310 is greater than 100, it is not included in this list, and further checks confirm it is not prime.

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 1310 as 2 × 655.

Step 2: In 2 × 655, 655 is a composite number. Further, break down 655 into 5 × 131.

Step 3: 131 is a prime number, so we now have the product consisting of only prime numbers. Hence, the prime factorization of 1310 is 2 × 5 × 131.

• 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.
   
• Twin primes: The pair of prime numbers whose difference is 2 are called twin prime numbers. For example, (11, 13) is a pair of twin primes because the difference between 13 and 11 is 2.
   
• 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.
   
• Prime numbers: Natural numbers greater than 1 with no divisors other than 1 and itself. For example, 5 is a prime number because its only divisors are 1 and 5.
   
• Divisibility: A number is divisible by another if it can be divided without leaving a remainder. For example, 10 is divisible by 2 because 10 ÷ 2 = 5 with no remainder.