Is 1280 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 1280 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 1280 has more than two factors, it is not a prime number. A few methods are used to distinguish between prime and composite numbers. Some 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 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 1280 is prime or composite.

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

Step 2: Divide 1280 by 2. It is divisible by 2, so 2 is a factor of 1280.

Step 3: Divide 1280 by 3. It is not divisible by 3, so 3 is not a factor of 1280.

Step 4: You can simplify checking divisors up to 1280 by finding the square root value. We then need to only check divisors up to the square root value.

Step 5: When we divide 1280 by 2, 4, 5, 8, and 10, it is divisible by 2, 4, 5, 8, and 10.

Since 1280 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 one's place value is 0. Zero is an even number, which means that 1280 is divisible by 2.

Divisibility by 3: The sum of the digits in the number 1280 is 11. Since 11 is not divisible by 3, 1280 is also not divisible by 3.

Divisibility by 5: The unit’s place digit is 0. Therefore, 1280 is divisible by 5.

Divisibility by 7: To check divisibility by 7, double the last digit (0 × 2 = 0). Then, subtract it from the rest of the number (128 - 0 = 128). Since 128 is not divisible by 7, 1280 is also not divisible by 7.

Divisibility by 11: In 1280, the sum of the digits in odd positions is 9, and the sum of the digits in even positions is 2. This would mean that 1280 is not divisible by 11.

Since 1280 is divisible by multiple numbers, it has more than two factors. Therefore, 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 the following steps.

Step 1: Write 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.

1280 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 1280 as 2 × 640.

Step 2: In 2 × 640, 640 is a composite number. Further, break the 640 into 2 × 320.

Step 3: Continue breaking down 320 into 2 × 160, then 160 into 2 × 80, 80 into 2 × 40, 40 into 2 × 20, and finally 20 into 2 × 10.

Step 4: Now, break down 10 into 2 × 5.

The prime factorization of 1280 is 2^8 × 5.

• 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 18 is 2 × 3².
   
• Divisibility test: A set of rules that determine if a number is divisible by another without performing division. For example, a number is divisible by 5 if its last digit is 0 or 5.
   
• Sieve of Eratosthenes: An ancient algorithm used to find all prime numbers up to a specified integer. It involves iteratively marking the multiples of each prime starting from 2.
   
• Co-prime numbers: Two numbers that have only 1 as their common factor. For example, 8 and 15 are co-prime because the only common factor is 1.