What are prime numbers & how to find prime numbers?

This is a GMAT Shots blog, aimed at providing GMAT aspirants with concise, useful information that’ll help with their GMAT Preparation. This blog deals with ‘What are Prime numbers, and how to determine if a number is prime?’

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What are prime numbers?

A Prime number put simply, is a positive integer that can only be divided by 1 and itself. A prime number has no other factors or divisors.

One important note – Prime numbers are a subset of natural numbers. So, when determining if a number is prime, we only talk about positive integers.

How to determine whether a given positive integer is a prime number?

There’s a simple 4-step method to determine if a given number ‘n’ is prime.

1. Find the closest perfect square that’s less than n.
2. Find the square root of the aforementioned perfect square.
3. List all prime numbers up to that square root.
4. Check divisibility of n by listed primes.

If any one of the listed primes divides our number ‘n’, then you know that ‘n’ is not a prime number.

Determining a Prime Number Example 1

Let’s take an example – the number 41.

1. The closest perfect square to 41 is 36.
2. The square root of 36 is 6.
3. The prime numbers up to 6 are 2, 3, and 5.
4. 41 is not divisible by any of 2, 3, or 5.

Because 41 is not divisible by any of the prime numbers listed in step 3, we can conclude that 41 is a prime number.

Prime Number Example 2

Let’s take another example – the number 39.

1. The closest perfect square to 39 is 36.
2. The square root of 36 is 6.
3. The prime numbers up to 6 are 2, 3, and 5.
4. The digits of 39 (3 and 9) add up to 12, and hence, 39 is divisible by 3.

Because 39 is divisible by one of the prime numbers listed in step 3, we can conclude that 39 is not a prime number.

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