Think like Einstein to solve Data Sufficiency

On the CAT we’ll want to reason more like Einstein, particularly when it comes to Data Sufficiency. In almost every class I teach, a student will ask a question along the lines of, “Is it possible that, in a value question, Statement 1 will tell you definitively that x equals 8, and that Statement 2 will tell you definitively that x equals some other number?” The answer is a resounding “No” – x has a unique value, the question is whether we can definitively divine what that value is. If Statement 1 tells us decisively that x = 8, Statement 2 cannot tell us that x equals, say, 10. Let’s see how this principle can be helpful in action: If a certain positive integer is divided by 9, the remainder is 3. What is the remainder when the integer is divided by 5? 1)     If the integer is divided by 45, the remainder is 30. 2)     The integer is divisible by 2 Statement 1 tells me that when I divide an integer by 45, I get a remainder of 30. So I could test 75, because that will give a remainder of 30 when divided by 45 (And, just as importantly, it gives a remainder of 3 when divided by 9 – I have to satisfy the conditions embedded in the question stem too!). The question asks me for the remainder when the integer is divided by 5. Well, 75/5 will give no remainder, so the remainder, in this case, is 0. Let’s see if that will always be the case. Next, we’ll test 105, which gives a remainder of 30 when divided by 45, and gives a remainder of 3 when divided by 9 [note: I can generate fresh numbers to test by simply adding the divisor, 30, to the previous number I test (75 + 30 = 105)]. Clearly 105/5 will give a remainder of 0, as any number that ends in 5 will be divisible by 5. The same will be true of 145, or 175, or 205. The remainder, when the integer in question is divided by 5 will always be 0, so Statement 1 is sufficient. Now let’s reason like Einstein. We know that the answer to the question has a definitive value of 0. That can’t change. The only way Statement 2 can be sufficient is if it gives us that same value. So let’s pick a number that is divisible by 2 but gives a remainder of 3 when divided by 9. 12 will work. The remainder, when 12 is divided by 5, is 2. All we need to see is that we did not get 0. We don’t have to test another number. Statement 2 cannot, alone, be sufficient, because we already know – the Einsteins that we are – that the value in question is 0. Statement 2 cannot tell us that the value is definitively 2 (if we continued to test, we’d eventually find values that gave us a remainder of 0 when we divided by 5, but because there are other possibilities, Statement 2 doesn’t give us enough information to determine, without a doubt, that the value is 0). We’re done. Statement 2 is insufficient. The answer is A: Statement 1 alone is sufficient. Note that this same logic will work on “YES/NO” questions as well. If Statement 1 tells us that the answer to the question is definitively “YES”, Statement 2 cannot tell us that the answer is definitively “NO”, and vice versa. Recognizing this can save us valuable time. Takeaway: For these DS questions we want to think more like Einstein and recognize that, in the mind of the question-writer, there is an objective answer – the question is whether we have enough information to definitely deduce what that answer is. There may be no objective reality in the quantum world, but on the CAT, there most certainly is.