Is 774 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 774 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 774 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 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 774 is prime or composite.

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

Step 2: Divide 774 by 2. It is divisible by 2, so 2 is a factor of 774.

Step 3: Divide 774 by 3. It is divisible by 3, so 3 is a factor of 774.

Step 4: You can simplify checking divisors up to 774 by finding the root value. We then need to check divisors up to the root value.

Step 5: When we divide 774 by 2, 3, and 5, it is divisible by 2, 3, and 5.

Since 774 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 4, an even number, which means that 774 is divisible by 2.

Divisibility by 3: The sum of the digits in the number 774 is 18. Since 18 is divisible by 3, 774 is also divisible by 3.

Divisibility by 5: The unit’s place digit is 4. Therefore, 774 is not divisible by 5.

Divisibility by 7: To check divisibility by 7, double the last digit (4 × 2 = 8). Then, subtract it from the rest of the number (77 - 8 = 69). Since 69 is not divisible by 7, 774 is also not divisible by 7.

Divisibility by 11: For 774, the alternating sum of the digits is 7 - 7 + 4 = 4, which means 774 is not divisible by 11.

Since 774 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, follow these steps:

Step 1: Write numbers from 1 to 1000 in rows and columns.

Step 2: Leave 1 without marking, 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 reaching the table consisting of marked and crossed boxes, except for 1. Through this process, we will have a list of prime numbers from 1 to 1000.

The list includes numbers like 2, 3, 5, 7, 11, 13, and so on. 774 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 774 as 2 × 387.

Step 2: In 387, which is a composite number, further break it down into 3 × 129.

Step 3: Now, decompose 129 into 3 × 43.

Step 4: 43 is a prime number.

Hence, the prime factorization of 774 is 2 × 3 × 3 × 43.

• 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 12 is divisible by 1, 2, 3, 4, 6, and 12.
   
• Prime factorization: Breaking down a number into its prime number components. For example, the prime factorization of 20 is 2 × 2 × 5.
   
• Divisibility test: A set of rules to determine if one number is divisible by another without performing division.
   
• Co-prime numbers: Two numbers with only 1 as their common factor.
   
• Even numbers: Numbers divisible by 2 without a remainder. For example, 4, 6, and 8 are even numbers.