This blog notes down all key concepts and properties to remember to be able to infer and use for prime number-related problems. Many GMAT Data Sufficiency problems will need the know-how of the properties so we can infer if sufficient information is available or not. GMAT prime numbers section is among the tricky part of the Quants.

**Prime Numbers Rules**

**Number 1:**1 is not a prime number. This is just to clarify in case, we need to count etc. So, 2 is the smallest prime number.**Divisibility**: A prime number is only divisible by**two distinct positive integers**1 and the number itself. A number is not prime if it’s divisible by any other positive integer.**Odd/Even:**All positive integers other 2 are odd by default. Also, this property is useful to know that**sum of two odds is even**,**so is the difference**. The**multiplication is not prime**either since both divide the number. The division is not possible.**Representation:**All prime numbers other than 2 and 3 can be represented as**6n-1**and**6n+1**, where n is a positive integer.**A number representable as 6n-1/6n+1 is not necessarily a prime number.****Goldbach Conjecture:**Any number other than**a sum of two prime numbers.****Consecutive Numbers:**2 and 3 are**only numbers that are consecutive integers and prime**. This is not necessarily true the other way around where 5 and 7 are prime but not consecutive integers.

### Is A Number **Prime or Not ? **

**One of the ways to check if a number is prime or not is to compute the closest square root of the number and jot down all prime numbers until then**. If the number is divisible by any of the prime numbers till then. The number then is not prime.

*e.g. 371, closest square root is sqrt(361) = 19. We can try dividing 371 with numbers from 2,3,5,7,11,13,17. If it’s divisible by any, it’s not a prime else it’s a prime number. The number indeed is divisible by 7, so not prime. *

But to speed up the process, we can try seeing if we can represent the **number as 6n-1 or 6n+1**. If not, we definitely know they are not prime and eliminate those answer choices. For those, which are valid, we need to confirm with the square root approach above. **A number representable as 6n-1/6n+1 gives remainder as 5 or 1 on being divided by 6.**

*371 in the above example gives a remainder as 5, so we went ahead and tried to divide it by different numbers. *

**Composite Concepts**

**Multiples:**A multiple is a number derived by multiplying the number a positive integer. This positive integer can go from 1 and beyond. Thereby,**the number itself is also multiple.****Factor:**A factor is a number that divides the number, it is the factor of.*eg. 2 is a factor of 20 and 20 is a multiple of 2.***A^2-B^2**: A^2-B^2 can be written as (A+B)*(A-B).

To be continued …