Is 1420 a Prime Number?

There are two main types of numbers—prime numbers and composite numbers—based on the number of factors they have.

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 have specific properties, such as:

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 because they have only one common factor, which is 1. As 1420 has more than two factors, it is not a prime number.

A prime number has only two divisors: 1 and itself. Since 1420 has more than two factors, it is not a prime number. Several methods are used to distinguish between prime and composite numbers, including:

• Counting Divisors Method

• Divisibility Test

• Prime Number Chart

• Prime Factorization

The counting divisors method involves counting the number of divisors to categorize numbers as prime or composite. Based on the count of the divisors, we categorize numbers:

If there is a total count of only 2 divisors, then the number is prime.

If the count is more than 2, then the number is composite. Let’s check whether 1420 is prime or composite.

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

Step 2: Divide 1420 by 2. It is divisible by 2, so 2 is a factor of 1420.

Step 3: Divide 1420 by 3. It is not divisible by 3, so 3 is not a factor of 1420.

Step 4: You can simplify checking divisors by finding the square root of 1420, which is approximately 37.7. We then need to check divisors up to this value.

Step 5: When we divide 1420 by 2, 5, and 7, it is divisible by 2 and 5. Since 1420 has more than 2 divisors, it is a composite number.

The divisibility test method involves using a set of rules to check if a number is divisible by another number completely.

Divisibility by 2: The number in the ones' place is 0, which means 1420 is divisible by 2.

Divisibility by 3: The sum of the digits in 1420 is 7, which is not divisible by 3, so 1420 is not divisible by 3.

Divisibility by 5: The unit’s place digit is 0, so 1420 is divisible by 5.

Divisibility by 7: Take 142, double the last digit (0 × 2 = 0), subtract it from the rest (142 - 0 = 142), and check if 142 is divisible by 7. It is not, so 1420 is not divisible by 7.

Divisibility by 11: The alternating sum of digits is 1 - 4 + 2 - 0 = -1, which is not divisible by 11. Since 1420 is divisible by 2 and 5, 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.” In this method, we follow the following steps:

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

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

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

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

Step 5: Repeat this process until you reach a table with marked and crossed boxes, except for 1. Through this process, we will have a list of prime numbers from 1 to 100. Since 1420 is not in the list of prime numbers, it is a composite number.

Prime factorization involves breaking down a number into prime factors and then multiplying those factors to obtain the original number.

Step 1: We can write 1420 as 2 × 710.

Step 2: Further break down 710 as 2 × 355.

Step 3: 355 can be broken down into 5 × 71.

Step 4: Now we have the product consisting of only prime numbers. Hence, the prime factorization of 1420 is 2 × 2 × 5 × 71.

• Composite numbers: Natural numbers greater than 1 that are divisible by more than 2 numbers. For example, 12 is a composite number because it is divisible by 1, 2, 3, 4, 6, and 12.

• Prime numbers: Natural numbers greater than 1 that are divisible by only 1 and itself. For example, 7 is a prime number because it is divisible only by 1 and 7.

• Divisibility: A condition where one number divides another without leaving a remainder.

• Prime factorization: The process of expressing a number as a product of its prime factors, such as 2 × 2 × 5 × 71 for 1420.

• Sieve of Eratosthenes: An ancient algorithm used to find all prime numbers up to a specified integer.