Prime numbers are special! In this chapter we will explore a key property of primes that can be quite useful for tricky GMAT Quant questions.

To start with: What is a prime number?

Essentially, a number greater than 1 which is only divisible by itself and 1.

2, 5, 7, 11, 13 and 157 are examples of prime numbers.

Let’s start with an example from data sufficiency:

**If x represents a positive integer, is y = (x + 3) ^{2} + 1 a prime number?**

**y is not divisible by any number <= (x + 3)****y is divisible by two number >= (x + 3)**

Readers are encouraged to take some time to solve the above!

**Prime square root theory: If z is not divisible by any number greater than 1 and <= √z, then it is a prime number. **

**This massively simplifies a lot of things! To check if a number is prime you do not have to test for divisibility for all numbers less than the number but by all numbers less than the square root of the number**

**To check if 1013 is a prime number you don’t have to test with ~1012 numbers, you just have to check with √1013 ~32 numbers. A 30 times magnitude simplification!**

Coming to the original question:

To test if y = (x+3)^{2} + 1 is a prime, we have to ensure all numbers greater than 1 and <= √y does not divide y.

Now √y = √((x+3)^{2} + 1) ~ (x + 3).

So, statement (1) alone is sufficient.

Statement (2) by itself confirms that y is not a prime. Given the two numbers >= (x+3)– both number have to be greater than 1 – since minimum value of x+3 is 4.

One of numbers has got to be y – then there is another number – which proves there is a factor besides y itself and 1.

So both statements individually are sufficient! We go with **option (D).**

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