Is 741 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.
   
• Since 741 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 741 has more than two factors, it is not a prime number. Several 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 741 is prime or composite.

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

Step 2: Divide 741 by 2. It is not divisible, so 2 is not a factor.

Step 3: Divide 741 by 3. The sum of the digits (7 + 4 + 1 = 12) is divisible by 3, so 3 is a factor.

Step 4: Divide 741 by 5. The last digit is not 0 or 5, so it is not divisible by 5.

Step 5: Further checks show 741 is divisible by 3, 13, and 19.

Since 741 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: 741 is odd, so it is not divisible by 2. 

Divisibility by 3: The sum of the digits (7 + 4 + 1 = 12) is divisible by 3, so 741 is divisible by 3. 

Divisibility by 5: The unit’s place digit is not 0 or 5, so it is not divisible by 5. 

Divisibility by 7: Applying the divisibility rule for 7, 741 is not divisible. 

Divisibility by 11: The alternating sum of the digits (7 - 4 + 1 = 4) is not divisible by 11, so it is not divisible by 11.

Since 741 is divisible by more than two numbers, 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 these 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, you will have a list of prime numbers from 1 to 1000. 741 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 741 as 3 × 247.

Step 2: In 3 × 247, 247 is a composite number. Further, break down 247 into 13 × 19.

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

Hence, the prime factorization of 741 is 3 × 13 × 19.

• 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: The process of breaking down a number into its prime factors. For example, 741 can be expressed as 3 × 13 × 19.

• Divisibility rules: Guidelines that help determine if one number is divisible by another without performing full division.

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

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