When the digits of two-digit, positive integer M are reversed, the res

Given: When the digits of two-digit, positive integer M are reversed, the result is the two-digit, positive integer N. Also, M > N
Let x = the tens digit of M
Let y = the units digit of M
So, 10x + y = the VALUE of M
When the digits are reversed, we get: 10y + x = the VALUE of N

So, M - N = (10x + y) - (10y + x) = 9x - 9y = 9(x - y)

Statement 1: The integer (M - N) has 12 unique factors.
M - N = 9(x - y)
So, statement 1 tells us that 9(x - y) has 12 unique factors.

------Aside---------------------
If the prime factorization of N = (p^a)(q^b)(r^c) . . . (where p, q, r, etc are different prime numbers), then N has a total of (a+1)(b+1)(c+1)(etc) positive divisors.

Example: 14000 = (2^4)(5^3)(7^1)
So, the number of positive divisors of 14000 = (4+1)(3+1)(1+1) =(5)(4)(2) = 40
--------------------------------

9(x - y) = 3²(x - y)
For the moment, notice that if (x-y) = (something)³, then M - N = 9(x - y) = 3²(something)³, in which case the number of positive divisors = (2+1)(3+1) =(3)(4) = 12
So, it must be the case that (x-y) = (something)³ (where "something" is a prime number)
Since x and y are non-zero DIGITS less than 10, it must be the case that (x-y) = 2³
In other words, it must be the case that x - y = 8

So, x = 9 and y = 1, which means M = 91 and N = 19
ASIDE: Notice that M - N = 91 - 19 = 72, and 72 has 12 positive factors.

So, the answer to the target question is M = 91
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The integer (M - N) is a multiple of 9.
As we've already seen from the work above, M - N will ALWAYS be a multiple of 9.
If you're not convinced, consider these two possible cases:
Case a: M = 41 and N = 14. Notice that M - N = 41 - 14 = 27, and 27 is a multiple of 9. In this case, the answer to the target question is M = 41
Case b: M = 31 and N = 13. Notice that M - N = 31 - 13 = 18, and 18 is a multiple of 9. In this case, the answer to the target question is M = 31
Since we can’t answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

I don't suggest you concern yourself with how long it takes to solve prep company questions. Often prep company questions are far more complicated than official questions, and the time they take is often irrelevant. That's true of this question - any question that requires a roughly 10-paragraph solution (like the one posted earlier in this thread) is not a realistic GMAT problem.

If you did want to solve: if M is the two-digit number AB, where A is the tens digit, B the units digit, then M = 10A + B. Since N has the same digits as M, but reversed, N is the number BA, so N = 10B + A. So

M - N = 10A + B - (10B + A) = 9A - 9B = 9(A - B)

We have learned that M-N is automatically a multiple of 9, so Statement 2 tells us nothing, and cannot possibly even be useful. So the answer must either be A or E.

Statement 1 is the time-consuming Statement. We know using that Statement that M-N has 12 divisors. That's a lot for a two-digit number, so there won't be many possible values of M-N. We also know M-N is divisible by 9, so we know it has at least 3^2 in its prime factorization. Now we need to imagine what the prime factorization of M-N could look like if M-N is going to have 12 divisors. We count a number's divisors by adding one to each exponent in the number's prime factorization, and multiplying the resulting numbers. We're looking for a two-digit number divisible by 9 with twelve divisors, and there are very few of those - if we imagine the smallest possible values M-N could have, being sure to include at least 3^2, we get the following possibilities:

72 = (2^3)(3^2) since this has 4*3 = 12 divisors
90 = (2^1)(3^2)(5^1) since this has 2*3*2 = 12 divisors

and any other possibility is larger than these (since we'd need to multiply larger primes - though you might want to check 2^2 * 3^3 to confirm it is too big). If M and N are both two digit numbers, M-N cannot be as large as 90 (the largest possible difference of two digit numbers is 89). So the only possibility is that M-N = 72, and since M-N = 9A - 9B, we learn that 9A - 9B = 72, so A - B = 8, and the digits of M are 8 apart. Since both digits are nonzero, and the tens digit of M is greater than the units digit, M can only be 91.

So Statement 1 is sufficient alone, but there's nothing remotely realistic about the question. It's not conceptually coherent - there's no relationship, conceptually speaking, between reversing digits and counting divisors, so the question is just haphazardly throwing unrelated mathematical ideas together, which is not how real GMAT questions are designed.

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

IanStewart has posted excellent detailed explanations above -- thanks Ian! I appreciate his note that unofficial problems are often more complicated and structured differently; I agree. They can definitely be a good challenge, but we shouldn't let them hurt our confidence too much -- it's most important to master the lower hanging fruit.

I've noticed the # of unique factors formula coming up much more often on unofficial problems, but I still find it useful on official problems occasionally (for example, here's one from the official CAT #1 & 2 question pool: https://gmatclub.com/forum/the-positive ... ml#p438422)

In case it's helpful for anyone, I'll post my whiteboard work for this below, with brief explanation added.

I'll also quote Bunuel's excellent explanation of the number of factors formula:

Hi NandishSS,

GMAT questions are always carefully worded and based on a pattern (and sometimes more than one). If you don't immediately see how to approach a question, you might shift your thinking over to the specific information that you are given and try to 'brute force' a solution based on those 'restrictions.'

Here, we know that M is a 2-digit number, N is the 'reverse' of that number and M > N so there are only a limited number of options for M and N...

We also know that M - N is ALSO a 2-digit number, which limits the options even more (for example, we can't have M=98 and N=89, since that difference is only 9 - which is a 1-digit number). Fact 1 tells us that (M - N) has 12 FACTORS.... and that is a LOT of factors for a number that is only 2 digits. Consider some simple examples...

(2)(5) = 10 and has 4 factors (1, 2, 5 and 10)
(2)(2)(5) = 20 and has 6 factors (1, 2, 4, 5, 10 and 20)

How do you get to 12 FACTORS while still having just two digits? Notice what happened when you included another "2" in the prime factorization? With a little playing around, you'll find that there aren't that many 2-digit numbers that have 12 factors AND fit all of the other information that you've been given... so how long with it take you to find them/it? (hint: it'll probably go a lot faster if you put your pen to the pad and got to work on some examples instead of rubbing your chin and staring at the ceiling).

Fact 2 is really easy to deal with, so you likely wouldn't spend much time on that piece of the question.

GMAT assassins aren't born, they're made,
Rich

A.

Let M = 10a+b
so N = 10b+a

(1) The integer (M - N) has 12 unique factors
M-N = 9(a-b) = 3^2(a-b)
3^2 already has 3 factors.
so (a-b) has 4 factors.
so (a-b) is of the form x*y or k^3 (where x, y, and k are prime numbers apart from 3)
(i) if (a-b) is of the form x*y then x,y can be 2,5,7,...
max difference between a and b is 8, which doesn't even satisfy the smallest product of 2*5. so this case is invalid.
(ii) if (a-b) is of the form k^3
note that (a-b) > 1 other wise M-N would only have 3 factors.
2^3 = 8 , 3^3 = 27 (not possible)
so (a-b) = 8
which is possible only for a=9,b=1.
hence, A alone is sufficient.

(2) The integer (M - N) is a multiple of 9
This is already known. Not useful...hence insufficient.

A good way to approach this problem is to look at the possible range of (M - N). The largest difference between digits would be 91 - 19 = 72, and the smallest would be something like 21-12 = 9. So you're working with a relatively limited range for the value of (M - N)
Express the number 72 as 2x2x2x3x3 = 2^3x3^2
Total number of factors can be found by ignoring base and taking only the exponents i.e 3 and 2. Add 1 to them and multiply.
i.e (3+1)x(2+1)= 4x3 = 12 factors available. If you use that in reverse here, you'll see that to get 12 as the total number of factors, you can have exponents of either 4 and 2 or 5 and 1. And since the smallest use of 5 and 1 would be 2^5∗3^1=96, which is out of the possible range, statement 1 must be sufficient.
Statement 2,is not sufficient. Note that you can express (A - B) algebraically using tens and units digits. For the two-digit number xy, for example, in which x and y are each digits (and not numbers to be multiplied), you'd algebraically say that the value is 10x + y, as x is the tens digit and y the units. That makes (A - B) equal to:
10x + y - (10y + x)
That simplifies to:
10x + y - 10y - x
Which is 9x - 9y, and can be expressed as 9(x - y). Statement 2 then doesn't tell us anything we don't already know; (A - B) MUST BE a multiple of 9, so we don't get any new information.

Statement 1:

I would show M as 10t + u and N as 10u + t. Where t is the tens digit of M and u is the units digit.

So 10t + u - 10u - t = 9t - 9u. We can factor out a 9 here 9(t-u) has 12 unique factors.

9 itself has a prime factorization of 3^2 so to find the number of unique factors of 9 we add 1 to the 2 (taken from 3^2) telling us 9 has 3 unique factors.

In order to get 12 unique factors we need a number to the power of 3.

The only number to the power of 3 when two single digits are subtracted from each other is 8, 2^3.

There (M-N) is equal to 3^2.2^3. Which as twelve unique factors 3.4 = 12.

Sufficient.

Statement 2:

Doesn't give us any new info.

Hi all,

I totally understand the logic, but what about the combination of the integers 8 and 0: 80-08=72 which has 12 factors.

It says nothing about non-zero digits...

Thanks in advance!

When the digits of two-digit, positive integer M are reversed, the result is the two-digit, positive integer N.

08 is just 8, so it's not a two-digit integer, it's a sing;e-digit integer.

Let \(M = 10a + b\), \(N = 10b +\)a

Statement 1: (M-N) has 12 unique factors.
\(M - N = 10a + b - 10b - a\)
\(= 9a - 9b = 9(a-b) = 3^2 * (a-b)\)
since \(M-N\) has 12 unique factors, \(3^2 *(a-b)\) must be of the form \(3^2 * prime^3\), so that total unique factors \((2+1) * (3+1) = 12\)
now '8' is only digit that fits \(prime^3\) which is \(2^3\).
so \((a-b) = 8\) => \(a = 9, b = 1\) OR\(a = 8, b = 0\)
but if a were 8 and b were 0, then reverse 08 won't be two digit positive integer (as prompted by the question)
so a must be 9 and b = 1, => M = 91 and N = 19 -> Sufficient

Statement 2: Doesn't tell us anything new, we already know from statement 1, \(M - N = 9(a-b)\) is multiple of 9
InSufficient.

Answer (A)

Let x = the tens digit and y = the units digit for M.
y cant be zero because when you reverse, the reversed digit will be single digit number and not a two digit number as required in the question
eg: M is represented as xy and value of M=10x+y
If y is zero and lets say x is 5
M=50 but, Mrev=05 which is not okay.
Hence y cannot be zero

Working on question stem first to extract max possible information before hitting the statements:
Since the statements provided have mentioned info about M-N lets break down M-N
M = 10x + y.
When the digits in M are reversed, we get Mrev which is N = 10y + x.

Difference, M-N = (10x + y) - (10y + x) = 9x - 9y = 9(x-y).

Max possible value of x-y = 9-1 = 8 (x and y are tens and units digit respectively, max possible value of x is 8 and min possible value of y is 1)

Thus, M-N = 9(x-y) can be:
9, 18, 27, 36, 45, 54, 63, 72 for (x-y)=1,2,3,4,6,5,6,7,8.
Thus we have all the possible values for M-N

Any statement alone/combination of both statements, if can pinpoint us to one of these values we have our answer, else option E.

S1:
(M - N) has 12 unique factors.
Of the options for M-N, only 72 has 12 unique factors:
72=(2^3)*(3^2)
Number of unique factors=(3+1)*(2+1)=4*3=12

Now, M-N = 9(x-y) = 72, we get:
9(x-y) = 72
x-y = 8, i.e x=9 and y=1
Result:
M = 10x + y = 10(9) + 1 = 91.

Thus S1 is sufficient ADBCE

S2:
The integer (M - N) is a multiple of 9.
We already know that because M-N=9(x-y) implies that M-N is a multiple of 9

S2 is insufficient. AD

Final answer: A

HI GMATGuruNY ,EMPOWERgmatRichC , @GMATPrepNow , IanStewart , MentorTutoring GMATCoachBen

Can you please help me with this problem, which is taking more than 2 mins...

really good question, Imo. stumped me completely at first. And then came the light bulb.

so\( M= xy \)(a two-digit number)
\(N= yx \)( two-digit number with digits reverses). one thing : \(M > N\)

here's how M looks like = 10x+y and N is = 10y+x

(1) The integer (M - N) has 12 unique factors.

\((10x+y) - (10y+x) \)-----> \(9x-9y \)----> \(9 (x-y)\). (I looked at this for at least a minute, not knowing how to proceed), and then. something I read somewhere on gmat club came back:

number of factors of a number for a number in the form : \(2^a* 3^b* 5^c= (a+1)*(b+1)*(c+1)\)
\(9 (x-y) = 3^2 (x-y)^ n\) --- (let say x-y has some power n, we don't know what)

so, \(3^2 (x-y)^ n\) ---> we know the # of factors = 12. therefore, \((2+1)* (n+1) =12 --- n = 3\)

now \((x-y) ^n = (x-y)^3\). therefore, we're left with \(9(x-y)^3\). if \(x-y = 3\), then \(3^3=27\) (which, when multiplied by 9 is not a 2-digit number. so, we reject this) thus. \((x-y)^3\) must be \((2)^3\) which when multiplied by 9 gives 72 <<<<<< M, therefore, \(N= 27\) ----> M> N. everything is satisfied.

!!!!phew. !!!!


(2) The integer (M - N) is a multiple of 9.

this just tells us (M-n) looks like : 9 (x-y)= so (M-N) can be any multiple of 9. clearly insufficient.

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