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A. 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).

That should get you thinking: 12 is a lot of factors. A number has to be relatively large just to have a large number of factors, so with a maximum value of 72 it's unlikely that more than a few number will have that. So start with 72. Its factors include:

1 and 72; 2 and 36; 3 and 24; 4 and 18; 6 and 12; 8 and 9. That's 12 total factors. And if you break it down into primes, it's 2 * 2 * 2 * 3 * 3, a combination of the two lowest prime factors available. For a smaller number to have as many factors, it doesn't have many options other than to turn those 3s into 2s. But try it using 2 * 2 * 2 * 2 * 3. That's 48, and 48 doesn't have 12 factors:

1 and 48; 2 and 24; 3 and 16; 4 and 12; 6 and 8. That's only 10 factors.

Because you can't find another difference that has 12 factors, M must be 91. And note that you can use the Unique Factors Trick to more quickly do the above:

1) Express the number as the product of prime numbers. (72 = 2 * 2 * 2 * 3 * 3)

2) Express that product using exponents for each prime base. (72=23∗32)

3) Forget about the bases and concentrate on the exponents (in this case 3 and 2)

4) Add one to each exponent (making them 4 and 3 in this case)

5) Multiply the exponents and you'll have your number of total factors.

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 25∗31=96, which is out of the possible range, statement 1 must be sufficient.

Statement 2, on the other hand, 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.

A. 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).

That should get you thinking: 12 is a lot of factors. A number has to be relatively large just to have a large number of factors, so with a maximum value of 72 it's unlikely that more than a few number will have that. So start with 72. Its factors include:

1 and 72; 2 and 36; 3 and 24; 4 and 18; 6 and 12; 8 and 9. That's 12 total factors. And if you break it down into primes, it's 2 * 2 * 2 * 3 * 3, a combination of the two lowest prime factors available. For a smaller number to have as many factors, it doesn't have many options other than to turn those 3s into 2s. But try it using 2 * 2 * 2 * 2 * 3. That's 48, and 48 doesn't have 12 factors:

1 and 48; 2 and 24; 3 and 16; 4 and 12; 6 and 8. That's only 10 factors.

Because you can't find another difference that has 12 factors, M must be 91. And note that you can use the Unique Factors Trick to more quickly do the above:

1) Express the number as the product of prime numbers. (72 = 2 * 2 * 2 * 3 * 3)

2) Express that product using exponents for each prime base. (72=23∗32)

3) Forget about the bases and concentrate on the exponents (in this case 3 and 2)

4) Add one to each exponent (making them 4 and 3 in this case)

5) Multiply the exponents and you'll have your number of total factors.

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 25∗31=96, which is out of the possible range, statement 1 must be sufficient.

Statement 2, on the other hand, 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.

The correct answer is A.

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.
_________________

Re: When the digits of two-digit, positive integer M are reversed, the res [#permalink]

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24 Oct 2014, 08:59

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This post was BOOKMARKED

From the premise, we already know that M-N is always a multiple of 9, so that immediately rules out statement 2 as insufficient. This is because when a positive 2 digit integer's digits are reverse, the difference between the two numbers is a multiple of 9.

I eventually got statement 1 to be sufficient. However, could someone clarify the rules with prime factorization--I'm still a bit confused.

A. 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).

That should get you thinking: 12 is a lot of factors. A number has to be relatively large just to have a large number of factors, so with a maximum value of 72 it's unlikely that more than a few number will have that. So start with 72. Its factors include:

1 and 72; 2 and 36; 3 and 24; 4 and 18; 6 and 12; 8 and 9. That's 12 total factors. And if you break it down into primes, it's 2 * 2 * 2 * 3 * 3, a combination of the two lowest prime factors available. For a smaller number to have as many factors, it doesn't have many options other than to turn those 3s into 2s. But try it using 2 * 2 * 2 * 2 * 3. That's 48, and 48 doesn't have 12 factors:

1 and 48; 2 and 24; 3 and 16; 4 and 12; 6 and 8. That's only 10 factors.

Because you can't find another difference that has 12 factors, M must be 91. And note that you can use the Unique Factors Trick to more quickly do the above:

1) Express the number as the product of prime numbers. (72 = 2 * 2 * 2 * 3 * 3)

2) Express that product using exponents for each prime base. (72=23∗32)

3) Forget about the bases and concentrate on the exponents (in this case 3 and 2)

4) Add one to each exponent (making them 4 and 3 in this case)

5) Multiply the exponents and you'll have your number of total factors.

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 25∗31=96, which is out of the possible range, statement 1 must be sufficient.

Statement 2, on the other hand, 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.

Could you explain how is the first point sufficient using a simpler solution.

For the second point, 'The integer (M - N) is a multiple of 9'

I followed the below logic.

two digit multiples of 9 are 18,27,36,45,54,63,72,81,90,99. Out of these 18,27,36,45 and 99 cannot be taken into consideration because it is given that M>N.

only possible considerations are 54,63,72,81,90. The only number satisfying the condition where M-N should be a multiple of 9 is 54.

M=54 and flipping the digits you get N=45.

M-N= 9 which is the only multiple possible. Am i missing anything here Bunuel. Need your help!!!!

A. 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).

That should get you thinking: 12 is a lot of factors. A number has to be relatively large just to have a large number of factors, so with a maximum value of 72 it's unlikely that more than a few number will have that. So start with 72. Its factors include:

1 and 72; 2 and 36; 3 and 24; 4 and 18; 6 and 12; 8 and 9. That's 12 total factors. And if you break it down into primes, it's 2 * 2 * 2 * 3 * 3, a combination of the two lowest prime factors available. For a smaller number to have as many factors, it doesn't have many options other than to turn those 3s into 2s. But try it using 2 * 2 * 2 * 2 * 3. That's 48, and 48 doesn't have 12 factors:

1 and 48; 2 and 24; 3 and 16; 4 and 12; 6 and 8. That's only 10 factors.

Because you can't find another difference that has 12 factors, M must be 91. And note that you can use the Unique Factors Trick to more quickly do the above:

1) Express the number as the product of prime numbers. (72 = 2 * 2 * 2 * 3 * 3)

2) Express that product using exponents for each prime base. (72=23∗32)

3) Forget about the bases and concentrate on the exponents (in this case 3 and 2)

4) Add one to each exponent (making them 4 and 3 in this case)

5) Multiply the exponents and you'll have your number of total factors.

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 25∗31=96, which is out of the possible range, statement 1 must be sufficient.

Statement 2, on the other hand, 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.

Could you explain how is the first point sufficient using a simpler solution.

For the second point, 'The integer (M - N) is a multiple of 9'

I followed the below logic.

two digit multiples of 9 are 18,27,36,45,54,63,72,81,90,99. Out of these 18,27,36,45 and 99 cannot be taken into consideration because it is given that M>N.

only possible considerations are 54,63,72,81,90. The only number satisfying the condition where M-N should be a multiple of 9 is 54.

M=54 and flipping the digits you get N=45.

M-N= 9 which is the only multiple possible. Am i missing anything here Bunuel. Need your help!!!!

Even 63 (&36) follows that M-N = multiple of 9.

63-36 = 27 = 9*3.

Thus you also have 63 as one of the possible answers.

A. 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).

That should get you thinking: 12 is a lot of factors. A number has to be relatively large just to have a large number of factors, so with a maximum value of 72 it's unlikely that more than a few number will have that. So start with 72. Its factors include:

1 and 72; 2 and 36; 3 and 24; 4 and 18; 6 and 12; 8 and 9. That's 12 total factors. And if you break it down into primes, it's 2 * 2 * 2 * 3 * 3, a combination of the two lowest prime factors available. For a smaller number to have as many factors, it doesn't have many options other than to turn those 3s into 2s. But try it using 2 * 2 * 2 * 2 * 3. That's 48, and 48 doesn't have 12 factors:

1 and 48; 2 and 24; 3 and 16; 4 and 12; 6 and 8. That's only 10 factors.

Because you can't find another difference that has 12 factors, M must be 91. And note that you can use the Unique Factors Trick to more quickly do the above:

1) Express the number as the product of prime numbers. (72 = 2 * 2 * 2 * 3 * 3)

2) Express that product using exponents for each prime base. (72=23∗32)

3) Forget about the bases and concentrate on the exponents (in this case 3 and 2)

4) Add one to each exponent (making them 4 and 3 in this case)

5) Multiply the exponents and you'll have your number of total factors.

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 25∗31=96, which is out of the possible range, statement 1 must be sufficient.

Statement 2, on the other hand, 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.

Could you explain how is the first point sufficient using a simpler solution.

For the second point, 'The integer (M - N) is a multiple of 9'

I followed the below logic.

two digit multiples of 9 are 18,27,36,45,54,63,72,81,90,99. Out of these 18,27,36,45 and 99 cannot be taken into consideration because it is given that M>N.

only possible considerations are 54,63,72,81,90. The only number satisfying the condition where M-N should be a multiple of 9 is 54.

M=54 and flipping the digits you get N=45.

M-N= 9 which is the only multiple possible. Am i missing anything here Bunuel. Need your help!!!!

Even 63 (&36) follows that M-N = multiple of 9.

63-36 = 27 = 9*3.

Thus you also have 63 as one of the possible answers.

Thanks Bunuel.. totally ignored that point. could you explain how the first point is sufficient. I did not understand the earlier solutions at all.

Re: When the digits of two-digit, positive integer M are reversed, the res [#permalink]

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23 Aug 2015, 19:52

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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.

Re: When the digits of two-digit, positive integer M are reversed, the res [#permalink]

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21 May 2016, 14:59

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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.

Re: When the digits of two-digit, positive integer M are reversed, the res [#permalink]

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02 Jan 2017, 23:36

[quote="pratikshr"]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.

let m = 10a+b and n = 10b+a , a>b , 2<=a<=9

from 1

9a-9b = m-n = 9(a-b) ,let (a-b) = k thus n = 3^2 *k and since m-n has 12 unique factors and since 9 has 3 unique factors (1,3,9) then k has 10 factors including 1.

a-b = can only yield ( 1 , 2,3,4,5,6,7,8)

number of factors of (3^2)*x^p is 3*(p+1) thus only a-b = 8 works since number of factors of 3^2 * 2^3 = 3*4 = 12

and since 2<=a<=9 thus a = 9 and b = 1 and thus m = 91 and N = 19..........suff

A. 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).

That should get you thinking: 12 is a lot of factors. A number has to be relatively large just to have a large number of factors, so with a maximum value of 72 it's unlikely that more than a few number will have that. So start with 72. Its factors include:

1 and 72; 2 and 36; 3 and 24; 4 and 18; 6 and 12; 8 and 9. That's 12 total factors. And if you break it down into primes, it's 2 * 2 * 2 * 3 * 3, a combination of the two lowest prime factors available. For a smaller number to have as many factors, it doesn't have many options other than to turn those 3s into 2s. But try it using 2 * 2 * 2 * 2 * 3. That's 48, and 48 doesn't have 12 factors:

1 and 48; 2 and 24; 3 and 16; 4 and 12; 6 and 8. That's only 10 factors.

Because you can't find another difference that has 12 factors, M must be 91. And note that you can use the Unique Factors Trick to more quickly do the above:

1) Express the number as the product of prime numbers. (72 = 2 * 2 * 2 * 3 * 3)

2) Express that product using exponents for each prime base. (72=23∗32)

3) Forget about the bases and concentrate on the exponents (in this case 3 and 2)

4) Add one to each exponent (making them 4 and 3 in this case)

5) Multiply the exponents and you'll have your number of total factors.

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 25∗31=96, which is out of the possible range, statement 1 must be sufficient.

Statement 2, on the other hand, 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.

The correct answer is A.

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.

"max difference between a and b is 8, which doesn't even satisfy the smallest product of 2*5. so this case is invalid. " thank you

Re: When the digits of two-digit, positive integer M are reversed, the res [#permalink]

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15 Apr 2017, 16:52

Bunuel wrote:

nachobs wrote:

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.

@bunuel- VeritasPrep gives the following solution; however, I think there is an error where state the possibilities for exponents are 5 and 1 and 4 and 2- wouldn't 4 and 2 be (4+1)(2+1)= 15- which is clearly not 12?

A. 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).

That should get you thinking: 12 is a lot of factors. A number has to be relatively large just to have a large number of factors, so with a maximum value of 72 it's unlikely that more than a few number will have that. So start with 72. Its factors include:

1 and 72; 2 and 36; 3 and 24; 4 and 18; 6 and 12; 8 and 9. That's 12 total factors. And if you break it down into primes, it's 2 * 2 * 2 * 3 * 3, a combination of the two lowest prime factors available. For a smaller number to have as many factors, it doesn't have many options other than to turn those 3s into 2s. But try it using 2 * 2 * 2 * 2 * 3. That's 48, and 48 doesn't have 12 factors:

1 and 48; 2 and 24; 3 and 16; 4 and 12; 6 and 8. That's only 10 factors.

Because you can't find another difference that has 12 factors, M must be 91. And note that you can use the Unique Factors Trick to more quickly do the above:

1) Express the number as the product of prime numbers. (72 = 2 * 2 * 2 * 3 * 3)

2) Express that product using exponents for each prime base. ( 72=23∗32 )

3) Forget about the bases and concentrate on the exponents (in this case 3 and 2)

4) Add one to each exponent (making them 4 and 3 in this case)

5) Multiply the exponents and you'll have your number of total factors.

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 25∗31=96 , which is out of the possible range, statement 1 must be sufficient.

Statement 2, on the other hand, 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.

Re: When the digits of two-digit, positive integer M are reversed, the res [#permalink]

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15 Apr 2017, 17:03

yezz 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.

let m = 10a+b and n = 10b+a , a>b , 2<=a<=9

from 1

9a-9b = m-n = 9(a-b) ,let (a-b) = k thus n = 3^2 *k and since m-n has 12 unique factors and since 9 has 3 unique factors (1,3,9) then k has 10 factors including 1.

a-b = can only yield ( 1 , 2,3,4,5,6,7,8)

number of factors of (3^2)*x^p is 3*(p+1) thus only a-b = 8 works since number of factors of 3^2 * 2^3 = 3*4 = 12

and since 2<=a<=9 thus a = 9 and b = 1 and thus m = 91 and N = 19..........suff

from 2

telling us what we already know ... insuff

A

But how can "K" have ten factors? If the answer is 3^2 x (a-b) then shouldn't k have 4 factors?