Prime Numbers – Properties & Applications

Is 1 a prime number or not? Can negative numbers be prime as well? How do I identify whether a given number is a prime number or not?

These are certainly some of the most common questions that a lot of us face, when it comes to Prime numbers. And it makes sense to know the answers to these questions and some more. Because, the concept of Prime numbers is tested extensively in many questions on GMAT Quant, and therefore, it becomes imperative that you know this concept well if you wish to score on those questions.

So let’s get started!

“Any positive integer, except 1, which has exactly TWO factors, 1 and itself, is called a prime number.”

Does this seem like too much to interpret in one go? Let’s then break it down further and understand the different parts of the definition.

1. First – For a number to be called a prime number, it has to be a positive integer.
2. Second – Of all the positive integers, the integer 1 does not qualify as a Prime number.
3. Third – It has to be divisible by exactly two factors – 1 and itself.

From the above analysis, it’s clear that negative numbers, ZERO, ONE, positive fractions and positive irrational numbers do not qualify as Prime numbers. Also, positive integers with MORE than 2 factors do not qualify as Prime numbers (because these are composite numbers)

This kind of an in-depth understanding of the basic property of a Prime number will pave the way for you to quickly identify whether a given number is prime or not.

Let us look at a few more important details about prime numbers.

The FIRST prime number is 2. It is also the ONLY EVEN prime number. This means that, if you exclude 2, any other prime number will ALWAYS BE ODD.

It’s a good idea to spend a few minutes in writing down the prime numbers from 1 to 100. This will familiarize you with these numbers and will put you in a good position to use them when the need arises. There are 25 prime numbers between 1 and 100. As you write down the prime numbers between 1 and 100, you will slowly observe that a pattern develops that can help you in identifying prime numbers. As we saw earlier, 2,3, 5 and 7 are the first four prime numbers. The next few prime numbers are – 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47.

What’s interesting is that, except for the first 4 prime numbers, you can observe that all other prime numbers always end with 1 or 3 or 7 or 9. But the contrary is not true always. What IS the contrary, you ask?

Any number ending with 1 or 3 or 7 or 9 need not necessarily be prime. Take the examples of 21, 33, 49 and 57. Although these numbers end with 1 or 3 or 7 or 9, these are not prime. Therefore, although the unit digit method can help us write down prime numbers quickly, it will not help us identify whether a given number is Prime.

Is there a more reliable technique which can be used to identify prime numbers? The answer is, YES.

Let's look at the number 5; 5 can be written as (6 * 1) – 1. Similarly, 7 can be written as (6*1) + 1; 11 can be written as (6*2) -1; 13 can be written as (6*2) + 1; 17 can be written as (6*3) – 1 and so on. So, any prime number can be written in the form of (6k) – 1 or (6k) + 1 where ‘k’ is a positive integer. But, wait. 2 cannot be written in either of the forms; 3 as well. Well, yes! These two numbers are the exceptions to this rule.

Any prime number greater than 3 can always be written in the form of (6k) -1 or (6k) + 1 where k is a positive integer.

Yet again, the vice versa is not always true. This means that any number in the form of (6k) -1 or (6k) + 1, k being a positive integer, need not necessarily be prime. Take the case of 25; 25 can be written as (6*4) + 1 but we know for a fact that 25 is not prime.

It’s therefore natural for you to have a question running in your mind at this point - “Is there a foolproof method to find out if a number is prime or not”. And not surprisingly, the answer is YES.

In the second part of this post, tomorrow, we will have a look at this method and also look at some related concepts on Numbers where Prime numbers are used.
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