Bunuel wrote:
If x is a positive, single-digit integer, is x prime?
(1) x^2 + 1 is prime.
(2) x + 18 is prime.
Kudos for a correct solution.
PRINCETON REVIEW OFFICIAL SOLUTION:Statement 1: INSUFFICIENT. To test the sufficiency of this statement, come up with values of x that satisfy the statement and then look upscreen to answer the question.
* For example, x = 1 satisfies the statement because 1^2 + 1 = 2, and 2 is prime. Now, look upscreen and go answer the question: Is x prime? If x = 1, x is not prime, so the answer is NO.
* However, x = 2 also satisfies the statement because 2^2 + 1 = 5, and 5 is prime. Again, look upscreen and answer the question: Is x prime? If x = 2, x is prime, so the answer is YES.
Now remember that if you can get both YES and NO answers to the question using the values that make a given statement true, as we did with Statement 1, that statement is insufficient: “Inconsistency = Insufficiency.”
Write BCE down on your scratch paper and go on to Statement 2 (you can write down BCE because if Statement 1 isn’t sufficient by itself, the answer can’t possibly be A or D).
Statement 2: INSUFFICIENT. Again, come up with values of x that satisfy the statement and then look upscreen to answer the question. However, this time try to recycle your work from Statement 1:
* First, check whether x = 1 satisfies Statement 2: If x = 1, x + 18 = 19, which is prime. Thus, x = 1 satisfies Statement 2, and you already know that you get a NO answer to the question when x = 1 (if x = 1, x is not prime, so the answer is NO).
* Next, check whether x = 2 satisfies Statement 2: If x = 2, x + 18 = 20, which is not prime. Thus, x = 2 does not satisfy Statement 2, so you won’t be able to use this value here.
* So far, you have only a NO answer (when x = 1), but though you couldn’t use x = 2 here, you still need to try to come up with a value for x that gives you a YES answer. In other words, you need to check whether there are any prime values of x that satisfy Statement 2.
* How about x = 3? If x = 3, x + 18 = 21, which is not prime, so you won’t be able to use x = 3 here.
* How about x = 5? If x = 5, x + 18 = 23, which is prime. Thus, x = 5 satisfies Statement 2. Again, look upscreen and answer the question: Is x prime? If x = 5, x is prime, so the answer is YES.
You now have both a YES answer (when x = 5) and a NO answer (when x = 1), so Statement 2 is insufficient.
Cross out B on your scratch paper and move on to checking whether both Statements together are sufficient to answer the question (if neither Statement is sufficient by itself, the answer can’t possibly be A, B, or D). You’re down to C vs. E.
Statements 1 & 2 Together: SUFFICIENT. Mine the overlap: x = 1 satisfies both Statements, and the answer is NO when x = 1. There were no other overlapping values between the two Statements. However, you still need to be sure that you’re not missing an overlapping value that gives you a YES answer to the question. To check, first try recycling from Statement 2 back to Statement 1:
* Does x = 5 satisfy Statement 1? If x = 5, x^2 + 1 = 26, which is not prime, so x = 5 does not satisfy Statement 1.
* You’ve now reached the moment of truth, when what you’ve learned from your sample values has to guide some thinking: For x to satisfy Statement 2 (x + 18 is prime), x has to be odd (not all odd values work, but even values won’t work because even + even = even, and 2 is the only even prime; furthermore, there’s no way you’ll get x + 18 = 2 because, don’t forget, the question told you that x has to be positive).
* Odd values other than x = 1 do not satisfy Statement 1 (x^2 + 1 is prime): An odd number squared will be odd (odd X odd = odd), and when you add 1 to an odd number, you get an even (odd + odd = even). But the only even prime is 2, and x^2 + 1 has to be prime for a particular value of x to satisfy Statement 1.
* Since only odd values satisfy Statement 2 and the only odd value that satisfies Statement 1 is x = 1, the only value that satisfies both Statements is x = 1.
Thus, there’s only one answer to the question when you combine Statements 1 & 2 (a definite NO), so Statements 1 & 2 Together are sufficient: “Consistency = Sufficiency.”
Circle answer C on your scratch paper, and you’re done with the question—nice job! (Remember, C means that neither Statement is sufficient by itself, but the Statements together are sufficient.)
The correct answer is (C).Our sample question shows that looking to recycle sample values between the two Statements and then mining the overlap is an essential, efficient strategy on Yes/No DS. There’s a lot to keep track of (especially when you try to recycle from Statement 2 back to Statement 1), so this will take plenty of practice. Also, be sure that you keep the fundamental process firmly in mind when you’re practicing: First, find values that satisfy the Statement, then look upscreen to answer the question YES or NO. Happy practice, and good luck!