Is 869 a Prime Number?

There are two main types of numbers — 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 869 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 869 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: -

1. Counting Divisors Method 
2. Divisibility Test 
3. Prime Number Chart 
4. 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 869 is prime or composite.

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

Step 2: Divide 869 by 2. It is not divisible by 2, so 2 is not a factor of 869.

Step 3: Divide 869 by 3. It is not divisible by 3, so 3 is not a factor of 869.

Step 4: You can simplify checking divisors up to 869 by finding the root value, which is approximately 29.5. We need to check divisibility only up to 29.

Step 5: When dividing 869 by numbers like 7, 11, or 13, it turns out that 869 is divisible by 11 (869 ÷ 11 = 79).

Since 869 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 9, which is odd. Therefore, 869 is not divisible by 2. 

Divisibility by 3: The sum of the digits in the number 869 is 8 + 6 + 9 = 23. Since 23 is not divisible by 3, 869 is also not divisible by 3. 

Divisibility by 5: The unit’s place digit is 9. Therefore, 869 is not divisible by 5. -

Divisibility by 7: The last digit is 9. Double it to get 18. Subtract this from the rest of the number (86 - 18 = 68). Since 68 is not divisible by 7, 869 is not divisible by 7. 

Divisibility by 11: The alternating sum of digits is (8 - 6 + 9 = 11), which is divisible by 11.

Since 869 is divisible by 11, it has more than two factors, making it 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 numbers from 1 to 100 in 10 rows and 10 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, we will have a list of prime numbers from 1 to 100.

The list is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97. Since 869 is greater than 100, we need to check against these and other factors, confirming it is not prime as it is divisible by 11.

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 869 as 11 × 79. Step 2: Both 11 and 79 are prime numbers.

Hence, the prime factorization of 869 is 11 × 79.

• 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 expressing a number as the product of its prime factors. For example, the prime factorization of 28 is 2 × 2 × 7. 

• Divisibility rules: Guidelines to 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. 

• Sieve of Eratosthenes: An ancient algorithm to find all prime numbers up to a specified integer. It systematically eliminates the multiples of each prime number starting from 2. 

• Co-prime numbers: Two numbers are co-prime if their greatest common divisor is 1. For example, 8 and 15 are co-prime.