There are two types of numbers, mostly —

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 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, 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 numbers because they have only one common factor, which is 1.
   
• Since 601 has exactly two factors, it is a prime number.

The characteristic of a prime number is that it has only two divisors: 1 and itself. Since 601 has no divisors other than 1 and 601, it is a prime number. Several methods can be used to distinguish between prime and composite numbers. A few methods are:

• 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 prime and composite numbers. 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 601 is prime or composite.

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

Step 2: Divide 601 by 2, 3, 4, 5, 6, 7... up to the square root of 601.

Step 3: Continue checking divisors until you find one or reach the square root of 601.

Since 601 is not divisible by any number other than 1 and itself, it is a prime 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: 601 is an odd number; hence, it is not divisible by 2.

Divisibility by 3: The sum of the digits in the number 601 is 7. Since 7 is not divisible by 3, 601 is also not divisible by 3.

Divisibility by 5: The unit’s place digit is 1. Therefore, 601 is not divisible by 5.

Divisibility by 7: Applying the divisibility rule for 7, 601 does not pass.

Since 601 is not divisible by any number other than 1 and itself, it is a prime 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 1000 in rows and 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 for all numbers up to the square root of 1000.

Through this process, we will have a list of prime numbers. Since 601 is not crossed out, it is a prime number.

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

Step 1: Attempt to factor 601 starting with the smallest prime number, 2.

Step 2: Continue with subsequent primes (3, 5, 7, 11...) until reaching the square root of 601.

Step 3: Since 601 cannot be divided by any prime number other than itself, it remains as a single prime factor, confirming that 601 is a prime number.

• Prime numbers: Natural numbers greater than 1 that have no divisors other than 1 and themselves. For example, 7 is a prime number.

• Composite numbers: Natural numbers greater than 1 that have more than two divisors. For example, 8 is a composite number because it is divisible by 1, 2, 4, and 8.

• Divisibility rules: A set of rules to determine whether one number is divisible by another without performing division. For example, a number is divisible by 3 if the sum of its digits is divisible by 3.

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

• Co-prime numbers: Two numbers that have no common factors other than 1. For example, 8 and 15 are co-prime.