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ETS Additional Test 1st edition, Section 5, #22

edited Thu Sep 14, 2017 1:54 am in Algebra
fontainedfontained Joined: 07/22/2017Posts: 23
(Admin note: The text of this question may be found in Section 5, Page 79 of the first edition of the GRE PDF practice book, archived here: https://web.archive.org/web/20110605194015/http://www.ets.org/s/gre/pdf/practice_book_GRE_pb_revised_general_test.pdf)

Could you please explain this problem?

Posts

  • edited Thu Sep 14, 2017 2:32 am
    Jonathan EvansJonathan Evans PowerScore Staff Joined: 10/31/2016Posts: 97
    Hi, Diane,

    Since we notice that there are variables in the answers and their values are not fixed (even though there are restrictions), this problem gives us a great opportunity to Supply Numbers.

    Let's establish how we would set up our scratch paper. Record What You Know:

    c, d : positive integers
    m : greatest common factor of c & d
    m : greatest common factor of c & what else?

    Just to keep the ball rolling, pick out some values for c & d. Perhaps we can try 2 & 3.

    c = 2
    d = 3

    What can we calculate now? The value of m:

    The greatest common factor of 2 and 3 is 1.

    Therefore m = 1.

    Let's use our values to ask the final question: "1 must be the greatest common factor of 2 and what else?"

    Use our values of c = 2 and d = 3 in the answer choices. Remember, we are looking for an answer that is the correct answer for the question, "1 must be the greatest common factor of 2 and what else?"

    A) 2 + 3 = 5 This works right now. 1 is the greatest common factor of 2 and 5, so we can leave it in.
    B ) 2 + 3 = 5 This works right now too. 1 is the greatest common factor of 2 and 5, so we can leave it in.
    C) 2 x 3 = 6 This doesn't work. 1 is not the greatest common factor of 2 and 6 (2 is), so we cross it out.
    D) 2 x 3 = 6 Same problem. Doesn't work. 1 is not the greatest common factor of 2 and 6, so we cross it out.
    E) 9 This works right now. 1 is the greatest common factor of 2 and 9, so we can leave it in.

    Using 2 and 3, we have narrowed down our answers to A, B, and E. If you're in a pinch, pick one of these three.

    However, to find the correct answer for sure we would need to supply different numbers. Notice that in this "must be true" type situation, we will likely have to supply numbers more than once, just as we do for Quantitative Comparison problems. 

    Also as with Quant Comp problems, the second time you supply numbers, you want to think about what you would want to supply to knock out remaining incorrect answers. The key is to be strategic. How could I pick different numbers that will cause the results to be different? Remember the list of numbers for Quant Comp: 1, -1, 2, -2, 0, 1/2, -1/2

    Because c and d are positive integers, some of these values don't work. However, there is nothing in the instructions that indicates that c and d have to be different from each other. In addition, since we saw 2 in the answer choices, let's stay away from that. Perhaps make both c and d 3. 

    c = 3, d = 3
    Now the greatest common factor of c and d is 3. m = 3

    Returning to our question, we ask: "3 must be the greatest common factor of 3 and what else?"

    We only need to look at A, B, and E this time.

    A) 3 + 3 = 6 This works right now. 3 is the greatest common factor of 3 and 6, so we can leave it in.
    B ) 2 + 3 = 5 This doesn't work. 3 is not the greatest common factor of 3 and 5 (1 is), so we cross it out.n.
    E) 9 This works right now. 3 is the greatest common factor of 3 and 9, so we can leave it in.

    Drat. We weren't able to get it down all the way to the right answer. Again, consider just making your best guess and moving on.

    To get to the right answer, we might have to change it up one more way. Let's do:

    c = 9, d = 3
    In this case, the greatest common factor of 9 and 3 is 3. m = 3

    Look only at A and E now. "3 must be the greatest common factor of 9 and what else?"

    A) 9 + 3 = 12 This works right now. 3 is the greatest common factor of 9 and 12, so we can leave it in.
    E) This doesn't work! 3 is not the greatest common factor of 9 and 9 (9 is), so we cross it out.

    We're done! A is the correct answer. 

    I understand this may still seem rather lengthy and difficult. It is indeed a hard problem. It presents you with a couple opportunities to "call it"—just say, "okay, that's enough. Time to make an educated guess."

    How could we guess correctly here? After the first round of elimination, you could reason the following:

    "2 doesn't have anything to do with anything. It's an arbitrary constant. Eliminate B."

    "Answer choice E doesn't even mention C. It seems likely that C would be relevant to getting the correct answer. Let's eliminate it as well."

    This is a perfectly valid approach to this problem. Remember, any method that efficiently gets you to the right answer is correct. I encourage you to attempt this question again to get the process down. 

  • fontainedfontained Joined: 07/22/2017Posts: 23
    Great thank you! Could you just eliminate D as well since it doesn't have a c in the expression? Or would this be bad to do right away?
  • Jonathan EvansJonathan Evans PowerScore Staff Joined: 10/31/2016Posts: 97
    That's reasonable and a good rule of thumb, but I actually wouldn't feel entirely comfortable with that. It works here, but there are situations in which the restrictions on multiple variables could make an answer that omits a variable work. 

    For instance, 

    "If c and d are positive even integers less than 5 and m is the least common multiple of c and d, then m must be a multiple of which of the following integers?"

    Either c² or d² would be acceptable for this question. 
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