Is 678 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 678 has more than two factors, it is not a prime number.

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The characteristic of a prime number is that it has only two divisors: 1 and itself. Since 678 has more than two factors, it is not 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 678 is prime or composite.

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

Step 2: Divide 678 by 2. It is divisible by 2, so 2 is a factor of 678.

Step 3: Divide 678 by 3. It is divisible by 3, so 3 is a factor of 678.

Step 4: You can simplify checking divisors up to 678 by finding the square root value. We then need to only check divisors up to the square root value.

Step 5: When we divide 678 by 2, 3, and further divisors up to its square root, we find it has multiple divisors.

Since 678 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: The number in the ones' place value is 8, which is an even number, meaning that 678 is divisible by 2.

Divisibility by 3: The sum of the digits in the number 678 is 21. Since 21 is divisible by 3, 678 is also divisible by 3.

Divisibility by 5: The unit’s place digit is 8. Therefore, 678 is not divisible by 5.

Divisibility by 7: The last digit in 678 is 8. To check divisibility by 7, double the last digit (8 × 2 = 16). Then, subtract it from the rest of the number (67 - 16 = 51). Since 51 is not divisible by 7, 678 is also not divisible by 7.

Divisibility by 11: In 678, the sum of the digits in odd positions is 14, and the sum of the digits in even positions is 7. Since the difference (14 - 7 = 7) is not divisible by 11, 678 is not divisible by 11. Since 678 is divisible by 2 and 3, 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 the following 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.

The list includes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and so on.

678 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 678 as 2 × 339.

Step 2: In 2 × 339, 339 is a composite number. Further, break the 339 into 3 × 113.

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

Hence, the prime factorization of 678 is 2 × 3 × 113.

• Composite numbers: Natural numbers greater than 1 that are divisible by more than 2 numbers are called composite numbers. For example, 678 is a composite number because it is divisible by 1, 2, 3, 6, 113, 226, 339, and 678.

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

• Divisibility rules: Guidelines that help determine if 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.

• Prime factorization: The process of expressing a number as a product of its prime factors. For example, the prime factorization of 30 is 2 × 3 × 5.

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