Is 73 a Prime Number

A prime number is always a positive integer that can be divisible by 1 and the number itself. These numbers can never be written a product of two numbers that are greater than 1. If any numbers other than 1 and the number itself divide it, there will always be a remainder. In the number system, other than prime numbers, there is composite number as well. These numbers have factors more than two.

73 is a prime number because it has only two factors, 1 and the number itself. To check whether a number like 73 is a prime number or not, we can use four different methods or ways to check it. They are: 

• Counting Divisor Method
• Divisibility Test
• Prime Number Chart
• Prime Factorization

The counting divisor method involves finding how many factors a number has to classify it as prime or composite. A number is categorized based on the total count of its factors.

Step 1: Identify the numbers to test for divisibility. You only need to test numbers from 2 up to the square root of 73. This is because any factor larger than the square root would require a smaller co-factor that would have already been tested.

The square root of 73 is approximately 8.54

2, 3, 5, and 7 are all prime numbers under 8.54.

Step 2: Check divisibility for each potential divisor.

For each number, you have to divide 73 by those numbers to check whether the quotient is an integer (without any remainder).

Test divisibility by 2, 4, 6, 8:

73 is odd, so it is not divisible by even numbers 2, 4, 6, and 8.

Test divisibility by 3:

73 divided by 3 gives 24 as the quotient and 1 as the remainder. So it is not a factor of 73.

Test divisibility by 5:

73 divided by 5 gives 14.6. So it is not a factor of 73.

Test divisibility by 7:

73 divided by 7 gives 10 as the quotient and 3 as the remainder. So it is not a factor of 73.

Step 3: Since 73 is not divisible by any number other than 1 and the number itself.
 

In the divisibility test method, we have to check whether a number is divisible by another without performing the actual division. These tests are based on the characteristics of numbers. They provide fast checks for divisibility by common factors, including 2, 3, 5, 7, 10, and others. 

Step 1: A number is divisible by 2 if the number ends with 0, 2, 4, 6, and 8    
 Since 73 is an odd number (it ends in 3), it is not divisible by 2.

Step 2: Check divisibility by 3
A number is divisible by 3 if the sum of its digits can be divided by 3.

Here, the sum of the digits of 73 is 7 + 3 = 10. Since 10 is not divisible by 3, 73 is not divisible by 3.

Step 3: Check divisibility by 5

A number is divisible by 5 if it ends with 0 or 5.

 Since 73 ends in 3, it is not divisible by 5.

Step 4: Check divisibility by 7

To check the divisibility by 7, divide 73 by 7.

73  7  10.43, which is not an integer.

 Since the result is not an integer, so 73 is not divisible by 7.

Step 5: Since 73 is not divisible by 2, 3, 5, and 7, it has no factors other than 1 and 73.

A Greek mathematician, Eratosthenes found a very interesting and easy method to find out prime numbers. It is called “The Sieve of Eratosthenes”. This chart will give you a better understanding of the prime numbers.

Step 1: Firstly, determine the range of numbers to check.

To use the Sieve of Eratosthenes, we first find the square root of 73. The square root of 73 is approximately 8.5, so we need to find all prime numbers up to 8. The prime numbers that are less than or equal to 8 are 2, 3, 5, or 7.

Step 2: Use the Sieve of Eratosthenes method 

Start with a list of numbers from 2 to 73. Like 2, 3, 4,5, 6, 7, 8, 9,.......,73.

We begin the process of eliminating non-prime numbers using the Sieve of Eratosthenes. In this method, we systematically remove all composite numbers by eliminating multiples of each prime number. 

Eliminate multiples of 2, starting with 4, 6, 8, 10, 12, and so on. Continue eliminating the multiples of 2. 

Eliminate multiples of 3, starting with 6, 9, 12, 15, and so on.

Eliminate multiples of 5, starting with 10, 15, 20, 25, and so on.

Eliminate multiples of 7, starting with 14, 21, 28, 35, and so on.
 
After performing these steps, the remaining numbers in the list will be the primes less than or equal to 73: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71.

Step 4: Check if 73 is in the list of primes

The number 73 appears in the list of primes, which means it is not eliminated in the Sieve of Eratosthenes.

Step 5: Since 73 is not divisible by any prime numbers less than or equal to 8, and it remains in the list of primes, 73 is a prime number.

• Prime Number: Numbers that are greater than 1 have only two factors: 1 and the number itself. For example, 3, 5, 7 are prime numbers. 

• Sieve of Eratosthenes: The easiest method introduced by a Greek mathematician to find all the prime numbers using a chart falling under a specific range.

• Square Root: The square root is a number, when multiplied by itself, gives the original number. For example, the square root of 16 is ±4, because 4  4 = 16.