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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