Is 420 a Prime Number?

There are two main types of numbers — prime numbers and 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 integer that has more than two factors. For example, 6 is divisible by 1, 2, 3, and 6, making it a composite number.

Prime numbers have certain properties, such as:

- Prime numbers are positive numbers 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.

Since 420 has more than two factors, it is not a prime number.

The defining characteristic of a prime number is that it has only two divisors: 1 and itself. Since 420 has more than two factors, it is not a prime number. Several methods can be used to identify whether a number is prime or composite, including:

• 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 is prime or composite.

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

- If it has more than 2 divisors, it is composite.

Let's check whether 420 is prime or composite.

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

Step 2: Divide 420 by 2. It is divisible by 2, so 2 is a factor of 420.

Step 3: Divide 420 by 3. It is divisible by 3, so 3 is a factor of 420.

Step 4: Continue dividing 420 by other numbers up to its square root.

Step 5: When we divide 420 by 2, 3, 5, 7, and others, we find multiple divisors.

Since 420 has more than 2 divisors, it is a composite number.

We use a set of rules to determine whether a number is divisible by another number without leaving a remainder. This is known as the Divisibility Test Method.

- Divisibility by 2: The last digit is 0, which is even, so 420 is divisible by 2.

- Divisibility by 3: The sum of the digits is 6 (4 + 2 + 0 = 6), which is divisible by 3, so 420 is divisible by 3.

- Divisibility by 5: The last digit is 0, so 420 is divisible by 5.

- Divisibility by 7: 420 divided by 7 leaves no remainder, so 420 is divisible by 7.

- Divisibility by 11: The alternating sum of digits is (4 - 2 + 0 = 2), which is not divisible by 11, so 420 is not divisible by 11.

Since 420 is divisible by several numbers, it has more than two factors, making it a composite number.

The prime number chart is a tool created using a method called “The Sieve of Eratosthenes.” This method involves:

Step 1: Writing numbers from 1 to 100 in 10 rows and 10 columns.

Step 2: Leaving 1 unmarked, as it is neither prime nor composite.

Step 3: Marking 2 as a prime number and crossing out all multiples of 2.

Step 4: Marking 3 as a prime number and crossing out all multiples of 3.

Step 5: Repeating this process until the chart is complete. By this method, we can identify the prime numbers between 1 and 100.

420 is not in the list of prime numbers, confirming it is composite.

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

Step 1: We can write 420 as 2 × 210.

Step 2: In 2 × 210, 210 is a composite number. Further, break 210 into 2 × 105.

Step 3: Break 105 into 3 × 35, and then break 35 into 5 × 7.

Step 4: Now we get the product of prime numbers 2 × 2 × 3 × 5 × 7.

Hence, the prime factorization of 420 is 2 × 2 × 3 × 5 × 7, showing it is a composite number.

• Composite numbers: Natural numbers greater than 1 that are divisible by more than 2 numbers are called composite numbers. For example, 8 is a composite number because it is divisible by 1, 2, 4, and 8.
  
   
• Divisibility rules: Guidelines that help determine if one number is divisible by another without performing full division.
  
   
• Prime factorization: The process of expressing a number as the product of its prime factors.
  
   
• Co-prime numbers: Two numbers that have no common factors other than 1. For example, 8 and 15 are co-prime.
  
   
• Sieve of Eratosthenes: An ancient algorithm used to find all prime numbers up to a specified integer.