Is 709 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 709 has exactly two factors, it is a prime number.

The characteristic of a prime number is that it has only two divisors: 1 and itself. Since 709 has exactly two factors, it is a prime number. A few 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 709 is prime or composite.

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

Step 2: Check divisibility by numbers up to the square root of 709, which is approximately 26.6.

Step 3: 709 is not divisible by any number other than 1 and 709 itself, confirming it has only two divisors.

Since 709 has exactly 2 divisors, it is a prime 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: 709 is not even, so it’s not divisible by 2.

Divisibility by 3: The sum of the digits in 709 is 16. Since 16 is not divisible by 3, 709 is also not divisible by 3.

Divisibility by 5: The unit’s place digit is 9, so 709 is not divisible by 5.

Divisibility by 7: Using the rule, 709 is not divisible by 7.

Divisibility by 11: Alternating sum and difference of digits give 2, which is not divisible by 11.

Since 709 is not divisible by any number other than 1 and itself, it is a prime 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, or use an available chart.

Step 2: Leave 1 without coloring or crossing, as it is neither prime nor composite.

Step 3: Mark known prime numbers like 2, 3, 5, 7, and cross out their multiples.

Step 4: Continue marking primes until all numbers have been evaluated. Checking a complete list of primes verifies that 709 is a prime number, as it is not crossed out and does not appear as a multiple of any other 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: Check divisibility of 709 by prime numbers up to its square root.

Step 2: 709 is not divisible by any prime numbers less than its square root.

Step 3: Since no prime factors are found other than 1 and 709 itself, 709 is a prime number.

• Prime Numbers: Natural numbers greater than 1 that are divisible by only 1 and themselves. For example, 11 is a prime number.

• Composite Numbers: Natural numbers greater than 1 that are divisible by more than 2 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 if a number is divisible by another without performing division, such as the rule for divisibility by 3.

• Perfect Square: A number that is the square of an integer. For example, 16 is a perfect square because it is 4 squared.

• Sieve of Eratosthenes: An ancient algorithm used to find all prime numbers up to any given limit, by iteratively marking the multiples of primes.