Is 674 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 674 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 674 has more than two factors, it is not a prime number. A few methods are used to distinguish between prime and composite numbers. These methods include:

• 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 674 is prime or composite.

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

Step 2: Divide 674 by 2. It is divisible by 2, so 2 is a factor of 674.

Step 3: Divide 674 by 3. It is not divisible by 3, so 3 is not a factor of 674.

Step 4: You can simplify checking divisors up to 674 by finding the square root value. We then need to only check divisors up to the root value.

Step 5: When we divide 674 by 2, it is divisible by 2. Since 674 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. This is called the Divisibility Test Method.

Divisibility by 2: The number in the ones' place value is 4. Four is an even number, which means that 674 is divisible by 2.

Divisibility by 3: The sum of the digits in the number 674 is 17. Since 17 is not divisible by 3, 674 is also not divisible by 3.

Divisibility by 5: The unit’s place digit is 4. Therefore, 674 is not divisible by 5.

Divisibility by 7: The last digit in 674 is 4. To check divisibility by 7, double the last digit (4 × 2 = 8). Then, subtract it from the rest of the number (67 - 8 = 59). Since 59 is not divisible by 7, 674 is not divisible by 7.

Divisibility by 11: In 674, the sum of the digits in odd positions is 10, and the sum of the digits in even positions is 7. The difference is 3, which is not divisible by 11. Since 674 is divisible only by 2 among the mentioned divisors, 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 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.

674 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 674 as 2 × 337.

Step 2: In 2 × 337, 337 is a prime number.

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

Hence, the prime factorization of 674 is 2 × 337.

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

• Divisibility rules: Guidelines that help determine whether one number is divisible by another. These rules help identify prime numbers.

• Prime factorization: The process of expressing a number as a product of its prime factors. Even numbers: Numbers divisible by 2 with no remainder.

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