NandishSS wrote:
pratikshr wrote:
When the digits of two-digit, positive integer M are reversed, the result is the two-digit, positive integer N. If M > N, what is the value of M?
(1) The integer (M - N) has 12 unique factors.
(2) The integer (M - N) is a multiple of 9.
HI
GMATGuruNY ,
EMPOWERgmatRichC , @GMATPrepNow ,
IanStewart ,
MentorTutoring GMATCoachBenCan you please help me with this problem, which is taking more than 2 mins...
Hello,
NandishSS. I also spent more than 2 minutes on this question, but I felt confident of my answer. (I sometimes spend up to 3 minutes to work through a question, since I know there will be others that, although they may be regarded as just as difficult, I will answer more quickly.) In this case, you have to appreciate the question stem.
Two digits result when integer M is reversed to become N, and M and N
cannot be the same number, since M > N. We can thus reduce our potential answer pool from all the integers 10-99, inclusive, to that same set minus any tens (10, 20, 30, and so on, since their reverse would become 01, 02, 03... single-digit numbers) and matching-digit integers (11, 22, 33, and so on). To be clear, when I was working on the problem, I did
not write down these values. I simply took a mental note and moved on.
That brings me to my second point. Whenever you are eyeing what appears to be a tough DS question, look to the statements for an easier point of entry. No matter how difficult one statement or the other may be to work through,
the two statements will always provide complementary information, even if one leads to a definitive answer and the other does not. Between (1) and (2), I would start with (2) any day of the week. If (M - N) is a multiple of 9, then the difference
must be 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, or 99... except that some of these values do not make sense. Consider:
1) The smallest difference between the integers M and N will be 9, when, for instance, M = 98 and N = 89, or M = 54 and N = 45.
2) The largest difference between the integers M and N will be 72, when M = 91 and N = 19.
How can we be certain of that second point? Because M = 99 is an invalid input, since N would equal 99 as well; M = 90 is an invalid input, since N would equal 9; any value for M less than 90 (e.g., 81) will only reduce the gap between M and N (e.g., 81 - 18 = 63, 82 - 28 = 54).
With this information in mind, we can eliminate any differences from before that are ruled out, and our list of potentials becomes 9, 18, 27, 36, 45, 54, 63, or 72. Now, once again, we can apply logic to finally see off this answer choice. I asked myself the following:
* Is there any reason M
has to be 91?
* Is there any reason M
cannot be 81, 72, etc.?
The answer to both is NO, so I cannot tell what the value of M may be.
Statement (2) is NOT sufficient. Get rid of answers (B) and (D) and get ready to tackle statement (1).
Okay, if (M - N) has 12 unique factors, then,
starting with my list from before, I can check for potential answers. The number we are looking for will clearly be larger to have that many factors. If we want, we can test 54:
1/54
2/27
3/18
6/9
Shucks. We hit a wall with 8 factors. How about 63?
1/63
3/21
7/9
Even worse. It should now be clear that we want an even number, since that will allow for more factors in which 2 and 3 can overlap. 72 seems a logical candidate:
1/72
2/36
3/24
4/18
6/12
8/9
Eureka! Now the question becomes, is 72 the only value that works? Since 2 and 3 seem to be prime candidates for checking multiples for overlap, we might want to consider numbers in the 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72 vein. Again, larger values make better potential targets. Since 12, 18, 24, and 36 are themselves factors of 72, we can eliminate them, since they cannot have the same number of factors as a larger multiple. Also, since we have already checked 72 (and 54), we can restrict the list further: 30, 42, 48, 60, 66. Keeping in mind that we cannot allow M to equal a number that ends in 0, we can also eliminate 30 and 60. There is simply no way to find a difference here that ends in a 0 without either M and N both equaling an even ten or having their units digit match, which we know cannot be true, since the two integers must be mirror images of each other and, again, M > N, not the same. The list now consists of just 42, 48, and 66. We can start with the largest:
1/66
2/33
3/22
6/11
Short again.
1/48
2/24
3/16
4/12
6/8
One pair shy of the mark.
1/42
2/21
3/4
6/7
Last one down. There are simply no other logical answers here.
M - N must equal 72, and that can only be true if M = 91 and N = 19, yielding a difference of 72. Since statement (1) allowed us to arrive at our conclusion,
(A) must be the answer.
If you were curious about other potentials, I did a little extra work (once I had chosen an answer) and figured out that any of 60, 72, 84, 90, and 96 all have 12 factors. Of course, all of them fail to meet the restrictions of the problem except for 72. Interesting, at least to me. (Maybe this information will come in handy on some other question.)
I hope that helps. Thank you for bringing my attention to the question.
- Andrew