Is 302 a Prime Number?

Numbers are generally categorized into two types —

Prime numbers and composite numbers, based on the number of factors they possess.

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 only 1 and 3.

A composite number is a positive number that has more than two divisors.

For example, 6 is divisible by 1, 2, 3, and 6, making it a composite number.

Prime numbers have specific properties such as:

• Prime numbers are 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 their only common factor is 1.
   
• Since 302 has more than two factors, it is not a prime number.

The defining characteristic of a prime number is that it has exactly two divisors: 1 and itself. Since 302 has more than two factors, it is not a prime number. There are several methods to determine whether a number is prime or composite:

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

The counting divisors method involves counting the number of divisors a number has to determine if it's prime or composite. Here's how it works:

If a number has exactly 2 divisors, it is prime.

If the number has more than 2 divisors, it is composite.

Let's check whether 302 is prime or composite.

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

Step 2: Divide 302 by 2. It is divisible by 2, so 2 is a factor of 302.

Step 3: Divide 302 by 3. It is not divisible by 3, so 3 is not a factor of 302.

Step 4: Check divisors up to the square root of 302 to simplify.

Since 302 is divisible by numbers other than 1 and itself, it is a composite number.

The divisibility test method involves using specific rules to check if a number is divisible by another without a remainder:

Divisibility by 2: The last digit of 302 is 2, which is even. Therefore, 302 is divisible by 2.

Divisibility by 3: The sum of the digits in 302 is 5. Since 5 is not divisible by 3, 302 is not divisible by 3.

Divisibility by 5: The last digit in 302 is not 0 or 5, so it is not divisible by 5. 

Divisibility by 7: Double the last digit (2 × 2 = 4) and subtract from the rest of the number (30 - 4 = 26). Since 26 is not divisible by 7, 302 is not divisible by 7.

Divisibility by 11: The difference between the sum of the odd-position digits and the even-position digits is 1 (3 - 0 + 2 = 5), so 302 is not divisible by 11.

Since 302 is divisible by factors other than 1 and 302, it is a composite number.

A prime number chart, created using the “Sieve of Eratosthenes” method, is a useful tool. Here are the steps:

Step 1: Write numbers from 1 to 100 in a 10x10 grid.

Step 2: Leave out 1 as it is neither prime nor composite.

Step 3: Mark 2 as a prime number and cross out all its multiples.

Step 4: Mark 3 as a prime number and cross out all its multiples.

Step 5: Continue this process for the next numbers until you complete the chart. The prime numbers up to 100 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Since 302 is not in this list, it is a composite number.

Prime factorization involves breaking down a number into its prime factors. Here’s how you would factorize 302:

Step 1: Divide 302 by 2 to get 151, as 2 is the smallest prime number that divides 302.

Step 2: 151 is a prime number, so the factorization stops here.

Thus, the prime factorization of 302 is 2 × 151.

• Composite Numbers: Natural numbers greater than 1 that have more than two factors. For example, 12 is composite because it is divisible by 1, 2, 3, 4, 6, and 12.
   
• Prime Numbers: Numbers greater than 1 with exactly two divisors, 1 and the number itself. For example, 7 is a prime number.
   
• Divisibility Rules: Guidelines used to determine if a number is divisible by another without performing the division.
   
• Sieve of Eratosthenes: An ancient algorithm used to find all prime numbers up to a specified integer.
   
• Prime Factorization: The expression of a number as the product of its prime factors. For example, 18 can be factorized into 2 × 3 × 3.