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 1061 has only two factors, it is a prime number.

The characteristic of a prime number is that it has only two divisors: 1 and itself. Since 1061 has no divisors other than 1 and 1061, it is a prime number. A few methods are used to distinguish between prime and composite numbers, including: 

• 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 numbers as prime or composite. - 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 1061 is prime or composite.

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

Step 2: Check divisibility of 1061 by numbers from 2 up to the square root of 1061.

Step 3: 1061 is not divisible by any number other than 1 and itself.
 

Since 1061 has exactly 2 divisors, 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: 1061 is odd, so it is not divisible by 2. 

Divisibility by 3: The sum of the digits in 1061 is 8. Since 8 is not divisible by 3, 1061 is not divisible by 3. 

Divisibility by 5: The unit’s place digit is 1, so 1061 is not divisible by 5. 

Divisibility by 7, 11, etc.: Continue this process for other primes up to the square root of 1061.

Since 1061 is not divisible by any smaller primes, it is a prime number.

The prime number chart is a tool created using a method called “The Sieve of Eratosthenes.” In this method, we follow these steps:

Step 1: Write numbers from 1 to 1000 (or more) in rows and columns.

Step 2: Leave 1 without coloring or crossing, as it is neither prime nor composite.

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

Step 4: Repeat this process for other numbers until you have a list of primes.

Through this process, we can determine that 1061 is a prime number as it remains unmarked.

Prime factorization is a process of breaking down a number into its prime factors. If a number can only be represented as a product of 1 and itself, it is prime. For 1061:

Step 1: Attempt to divide 1061 by primes less than its square root (approximately 32.6).

Step 2: 1061 is not divisible by any primes up to 32.

Hence, 1061 cannot be broken down further and is a prime number.

• Prime numbers: Natural numbers greater than 1 that have no divisors other than 1 and themselves. For example, 11 is a prime number.

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

• Divisibility test: A method to determine if one number is divisible by another without performing division.

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

• Factors: The numbers that divide into another number exactly without leaving a remainder. For example, the factors of 6 are 1, 2, 3, and 6.