Is 677 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 that is 1.

As 677 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 677 has only two factors, it is a prime number. Few methods are 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 677 is prime or composite.

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

Step 2: Check divisibility of 677 by prime numbers up to the square root of 677.

Step 3: 677 is not divisible by 2, 3, 5, 7, 11, 13, 17, 19, or other primes up to its square root.

Since 677 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: 677 is an odd number, so it is not divisible by 2.

Divisibility by 3: The sum of the digits in 677 is 20. Since 20 is not divisible by 3, 677 is not divisible by 3.

Divisibility by 5: The unit’s place digit is 7. Therefore, 677 is not divisible by 5.

Divisibility by 7, 11, 13, etc.: Check using calculations, none of these prime numbers divide 677 completely. Since 677 is not divisible by any prime numbers 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 numbers in a sequence.

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

Step 3: Mark the smallest prime number and cross out all its multiples.

Step 4: Repeat this process until all numbers have been marked or crossed. Using this method, we find that 677 is not crossed out, indicating 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 break down 677 into two or more factors.

Step 2: 677 cannot be divided by any smaller prime numbers. Since 677 cannot be factored into smaller prime numbers, it is a prime number.

• Prime numbers: Natural numbers greater than 1 that have no divisors other than 1 and themselves.

• Composite numbers: Natural numbers greater than 1 that have more than two divisors.

• Divisibility: A concept in which one number can be divided by another without leaving a remainder.

• Sieve of Eratosthenes: A simple algorithm to find all prime numbers up to a specified integer.

• Factors: The numbers that divide another number exactly without leaving a remainder.