109 LearnersLast updated on May 26th, 2025
Is 420 a Prime Number?
The numbers that have only two factors, which are 1 and themselves, are called prime numbers. They are fundamental in cryptography, computer algorithms, and other applications. In this topic, we will be discussing whether 420 is a prime number or not.
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.
Why is 420 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
Using the Counting Divisors Method
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.
Using the Divisibility Test Method
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.
Using Prime Number Chart
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.
Using the Prime Factorization Method
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.
Common Mistakes to Avoid When Determining if 420 is Not a Prime Number
There are common misconceptions about prime numbers when learning about them. Here are some mistakes that might occur.
Mistake 1
Assuming That All Odd Numbers Are Prime
Solution: It is often assumed that all odd numbers are prime. However, except for 2, most prime numbers are odd, but not all odd numbers are prime.
For example, 9 is an odd number but has more than two factors, making it composite.
FAQ on is 420 a Prime Number?
1.Is 420 a perfect square?
2.What is the sum of the divisors of 420?
3.What are the factors of 420?
4.What are the closest prime numbers to 420?
5.What is the prime factorization of 420?
Important Glossaries for "Is 420 a Prime 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.
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Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
