Is 777 a Prime Number?

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 like: 

• 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.
   
• As 777 has more than two factors, it is not a prime number.

The characteristic of a prime number is that it has only two divisors: 1 and itself. Since 777 has more than two factors, it is not a prime number. Several methods can be used to differentiate 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 numbers as prime or composite. - 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 777 is prime or composite.

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

Step 2: Divide 777 by 2. It is not divisible by 2, as it’s an odd number.

Step 3: Divide 777 by 3. The sum of its digits (7 + 7 + 7) is 21, which is divisible by 3, so 3 is a factor of 777.

Step 4: You can simplify checking divisors up to 777 by finding the root value. We then need to only check divisors up to the root value.

Step 5: Continuing with divisors such as 5, 7, and others, we find more factors.

Since 777 has more than 2 divisors, it is a composite 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: 777 is an odd number, so it is not divisible by 2. 

Divisibility by 3: The sum of the digits in the number 777 is 21. Since 21 is divisible by 3, 777 is also divisible by 3. 

Divisibility by 5: The unit’s place digit is 7. Therefore, 777 is not divisible by 5. 

Divisibility by 7: Double the last digit (7 × 2 = 14) and subtract it from the rest of the number (77 - 14 = 63). Since 63 is divisible by 7, 777 is also divisible by 7. 

Divisibility by 11: Calculate the difference between the sum of the digits in odd and even positions. (7 + 7) - 7 = 7. Since 7 is not divisible by 11, 777 is not divisible by 11.

Since 777 is divisible by 3 and 7, it has more than two factors. Therefore, it is a composite number.

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

Step 1: Write numbers from 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 until you reach the table consisting of marked and crossed boxes, except 1. Through this process, we will have a list of prime numbers from 1 to 1000.

777 is not present in the list of prime numbers, so it is a composite 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: We can write 777 as 7 × 111.

Step 2: In 7 × 111, 111 is a composite number. Further, break 111 into 3 × 37.

Step 3: Now we get the product consisting of only prime numbers.

Hence, the prime factorization of 777 is 3 × 7 × 37.

• Prime number: A natural number greater than 1 that is not divisible by any number other than 1 and itself.
   
• Composite number: A natural number greater than 1 that is divisible by more than two numbers.
   
• Divisibility rules: A set of rules that help determine whether one number is divisible by another without a remainder.
   
• Factors: The numbers that divide a given number exactly without leaving a remainder.
   
• Prime factorization: The process of expressing a number as the product of its prime factors.