Is 738 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 738 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 738 has more than two factors, it is not a prime number. A few methods are used to distinguish between prime and composite numbers, such as: 

• 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 738 is prime or composite.

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

Step 2: Divide 738 by 2. It is divisible by 2, so 2 is a factor of 738.

Step 3: Divide 738 by 3. It is divisible by 3, so 3 is a factor of 738.

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

Since 738 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 8. Since 8 is an even number, 738 is divisible by 2. 

Divisibility by 3: The sum of the digits in the number 738 is 18. Since 18 is divisible by 3, 738 is also divisible by 3. 

Divisibility by 5: The unit’s place digit is 8. Therefore, 738 is not divisible by 5. 

Divisibility by 7: To check divisibility by 7, double the last digit (8 × 2 = 16). Then, subtract it from the rest of the number (73 - 16 = 57). Since 57 is divisible by 7, 738 is also divisible by 7. 

Divisibility by 11: For 738, the alternating sum is 7 - 3 + 8 = 12, which is not divisible by 11, so 738 is not divisible by 11.

Since 738 is divisible by more than two numbers, it is a composite number.

A prime number chart is a tool created 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 desired range.

Through this process, we will have a list of prime numbers. 738 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 738 as 2 × 369.

Step 2: In 2 × 369, 369 is a composite number. Further, break down 369 into 3 × 123.

Step 3: Now, break down 123 into 3 × 41.

Thus, the prime factorization of 738 is 2 × 3 × 3 × 41.

•  Composite numbers: Natural numbers greater than 1 that are divisible by more than two numbers are called composite numbers. For example, 14 is a composite number because 14 is divisible by 1, 2, 7, and 14. 

• Divisibility rules: A set of rules that help determine whether a 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. 

• Factors: The numbers that divide the number exactly without leaving a remainder are called factors. For example, the factors of 8 are 1, 2, 4, and 8 because they divide 8 completely. 

• Prime factorization: The process of expressing a number as a product of its prime factors. For example, the prime factorization of 18 is 2 × 3 × 3. 

• Sieve of Eratosthenes: An ancient algorithm used to find all primes up to a specified integer. It involves iteratively marking the multiples of primes starting from 2.