Data Sufficiency

If x and y are positive integers such that x²+y³ is a prime number less than 18, what is the value of y?

1. x²+y² is a prime number
2. x²-y² is a prime number

This is a pretty time-consuming question.

Target question: What is the value of n?

Given: x and y are positive integers such that x²+y³ is a prime number less than 18

Since x²+y³ must be less than 18, the values of x and y are quite restricted.
x can equal 1, 2, 3 or 4
y can equal 1 or 2

Let's first check all 8 possible configurations of x²+y³ to see which ones are prime:
x = 1 and y = 1: x²+y³ = 1 + 1 = 2 (PRIME)
x = 1 and y = 2: x²+y³ = 1 + 8 = 9 (NOT prime)
x = 2 and y = 1: x²+y³ = 4 + 1 = 5 (PRIME)
x = 2 and y = 2: x²+y³ = 4 + 8 = 12 (NOT prime)
x = 3 and y = 1: x²+y³ = 9 + 1 = 2 (NOT prime)
x = 3 and y = 2: x²+y³ = 9 + 8 = 17 (PRIME)
x = 4 and y = 1: x²+y³ = 16 + 8 = 17 (PRIME)
x = 4 and y = 2: x²+y³ = 16 + 8 = 24 (NOT prime and too big)

So, there are 4 possible cases where x²+y³ is prime. They are:
case a) x = 1 and y = 1
case b) x = 2 and y = 1
case c) x = 3 and y = 2
case d) x = 4 and y = 1

Now check the statements:

Statement 1: x²+y² is a prime number
Let's test all 4 possible cases:
case a) x = 1 and y = 1: Here x²+y² = 1+1 = 2 (PRIME). In this case y = 1
case b) x = 2 and y = 1: Here x²+y² = 4+1 = 5 (PRIME). In this case y = 1
case c) x = 3 and y = 2: Here x²+y² = 9+4 = 13 (PRIME). In this case y = 2
case d) x = 4 and y = 1: Here x²+y² = 16+1 = 17 (PRIME). In this case y = 1
Since y can equal 1 or 2, statement 1 is NOT SUFFICIENT

Statement 2: x²-y² is a prime number
Let's test all 4 possible cases:
case a) x = 1 and y = 1: Here x²-y² = 1-1 = 0 (NOT prime). So, case a is not possible.
case b) x = 2 and y = 1: Here x²-y² = 4-1 = 3 (PRIME). In this case y = 1
case c) x = 3 and y = 2: Here x²-y² = 9-4 = 5 (PRIME). In this case y = 2
case d) x = 4 and y = 1: Here x²-y² = 16-1 = 15 (NOT prime). So, case d is not possible.
Since y can equal 1 or 2, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that cases a, b, c and d are all possible
Statement 2 tells us that cases b and c are possible
So, when the statements are combined, we can conclude that cases b and c are possible.
Case b tells us that y = 1
Case c tells us that y = 2
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer = E

Cheers,
Brent