Is 767 a Prime Number?

Numbers can be classified into two types —

Prime numbers and composite numbers, based on the number of factors they have.

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 have a few properties: 

• 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 767 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 767 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 767 is prime or composite. 

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

Step 2: Divide 767 by 2. It is not divisible by 2, so 2 is not a factor of 767. 

Step 3: Divide 767 by 3. The sum of its digits is 20, which is not divisible by 3, so 3 is not a factor of 767. 

Step 4: Continue checking divisibility by prime numbers up to the square root of 767.

Since 767 can be divided by 13 (767 ÷ 13 = 59), it is a composite number because it has more than 2 divisors.

We use a set of rules to check whether a number is divisible by another number completely. This is called the Divisibility Test Method. 

Divisibility by 2: 767 is an odd number, so it is not divisible by 2. 

Divisibility by 3: The sum of the digits in 767 is 20. Since 20 is not divisible by 3, 767 is not divisible by 3. 

Divisibility by 5: The unit’s place digit is 7. Therefore, 767 is not divisible by 5. 

Divisibility by 7: Double the last digit (7 × 2 = 14) and subtract from the rest (76 - 14 = 62). Since 62 is not divisible by 7, 767 is not divisible by 7. 

Divisibility by 11: The difference between the sum of the digits in odd and even positions is 6 - 7 = -1, which is not divisible by 11, so 767 is not divisible by 11.

Since 767 is divisible by 13, it has more than two factors, confirming it is a composite number.

The 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 unmarked, as it is neither prime nor composite. 

Step 3: Mark 2 as a prime number and cross out all multiples of 2. 

Step 4: Mark 3 as a prime number and cross out all multiples of 3. 

Step 5: Repeat this process for other primes. By following this method, we can identify prime numbers within a range.

Since 767 does not appear on the list of prime numbers, it is a composite number.

Prime factorization is the process of breaking down a number into its prime factors and then multiplying those factors to obtain the original number. 

Step 1: Start with the smallest prime number that divides 767, which is 13. 

Step 2: Divide 767 by 13 to get 59.

Step 3: Both 13 and 59 are prime numbers, so the prime factorization of 767 is 13 × 59.

• 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 expression of a number as a product of its prime factors. For instance, the prime factorization of 12 is 2 × 2 × 3.
   
• Divisibility rules: Guidelines that help determine if one number is divisible by another without performing division.
   
• Co-prime numbers: Two numbers are co-prime if they have no common factors other than 1. For example, 15 and 28 are co-prime numbers.
   
• Sieve of Eratosthenes: An ancient algorithm used to find all prime numbers up to a specified integer.