It is currently 20 Nov 2017, 18:16

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

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

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

3 KUDOS received
Intern
Intern
avatar
Joined: 05 Nov 2013
Posts: 24

Kudos [?]: 87 [3], given: 69

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

Show Tags

New post 30 Sep 2014, 06:56
3
This post received
KUDOS
31
This post was
BOOKMARKED
00:00
A
B
C
D
E

Difficulty:

  85% (hard)

Question Stats:

52% (01:13) correct 48% (01:16) wrong based on 386 sessions

HideShow timer Statistics

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.

Official Explanation:
[Reveal] Spoiler:
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.
[Reveal] Spoiler: OA

Kudos [?]: 87 [3], given: 69

Intern
Intern
avatar
Joined: 05 Nov 2013
Posts: 24

Kudos [?]: 87 [0], given: 69

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

Show Tags

New post 30 Sep 2014, 06:56
Did not quite get the hang of the provided OA.

Can a math expert please provide an alternative approach?

Kudos [?]: 87 [0], given: 69

14 KUDOS received
Intern
Intern
avatar
Joined: 23 Aug 2014
Posts: 8

Kudos [?]: 26 [14], given: 1

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

Show Tags

New post 23 Oct 2014, 16:11
14
This post received
KUDOS
9
This post was
BOOKMARKED
I solved the question like this :
As per the give info , M = 10a + b ; N =10b + a
Also, M>N

Lets analyse statements now:
Stmt 1 : The integer (M - N) has 12 unique factors.

M - N = (10a + b) - (10b + a) = 9a - 9b
so, M - N = 9 (a-b) = 3^2 * (a-b)

Stmt1 says M - N has 12 factors , this implies that (a-b) should be a number with power as 3.

Remember the rule , if prime factorization of the integer ,N = X^p * Y^q * Z^r , then the number of factors of N = (p+1)*(q+1)*(r+1)

So, a-b should be a number with power as 3. This implies it should be 8 , which is 2^3.

Hence, (a - b) = 8 , this implies a= 9 and b =1

So, M= 91 and N = 19 , coz given is M>N

Stmt1: Sufficient

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

M - N = (10a + b) - (10b + a) = 9a - 9b
so, M - N = 9 (a-b) . This is already a multiple of 9.

Therefore, (a-b) can be any integer. Hence we cannot narrow down the values of a & b to find M.

Stmt2 : Insufficient

Kudos [?]: 26 [14], given: 1

1 KUDOS received
Manager
Manager
User avatar
B
Joined: 22 Jan 2014
Posts: 141

Kudos [?]: 77 [1], given: 145

WE: Project Management (Computer Hardware)
Re: When the digits of two-digit, positive integer M are reversed, the res [#permalink]

Show Tags

New post 24 Oct 2014, 06:39
1
This post received
KUDOS
2
This post was
BOOKMARKED
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.

Official Explanation:
[Reveal] Spoiler:
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.
_________________

Illegitimi non carborundum.

Kudos [?]: 77 [1], given: 145

Manager
Manager
avatar
Joined: 12 Sep 2014
Posts: 168

Kudos [?]: 82 [0], given: 103

Concentration: Strategy, Leadership
GMAT 1: 740 Q49 V41
GPA: 3.94
Re: When the digits of two-digit, positive integer M are reversed, the res [#permalink]

Show Tags

New post 24 Oct 2014, 08:59
1
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.

Kudos [?]: 82 [0], given: 103

Intern
Intern
avatar
Joined: 25 Jan 2014
Posts: 17

Kudos [?]: [0], given: 75

Concentration: Technology, General Management
Premium Member
Re: When the digits of two-digit, positive integer M are reversed, the res [#permalink]

Show Tags

New post 03 Aug 2015, 13:27
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.

Official Explanation:
[Reveal] Spoiler:
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.


Bunuel,

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!!!!

Kudos [?]: [0], given: 75

Current Student
avatar
B
Joined: 20 Mar 2014
Posts: 2676

Kudos [?]: 1773 [0], given: 794

Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44
GPA: 3.7
WE: Engineering (Aerospace and Defense)
GMAT ToolKit User Premium Member Reviews Badge
Re: When the digits of two-digit, positive integer M are reversed, the res [#permalink]

Show Tags

New post 03 Aug 2015, 13:53
arshu27 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.

Official Explanation:
[Reveal] Spoiler:
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.


Bunuel,

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.

Kudos [?]: 1773 [0], given: 794

Intern
Intern
avatar
Joined: 25 Jan 2014
Posts: 17

Kudos [?]: [0], given: 75

Concentration: Technology, General Management
Premium Member
Re: When the digits of two-digit, positive integer M are reversed, the res [#permalink]

Show Tags

New post 03 Aug 2015, 13:58
Engr2012 wrote:
arshu27 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.

Official Explanation:
[Reveal] Spoiler:
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.


Bunuel,

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.

Kudos [?]: [0], given: 75

Intern
Intern
avatar
Joined: 02 Jan 2015
Posts: 21

Kudos [?]: 15 [0], given: 15

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

Show Tags

New post 23 Aug 2015, 19:52
1
This post was
BOOKMARKED
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.

Kudos [?]: 15 [0], given: 15

2 KUDOS received
Intern
Intern
avatar
B
Joined: 20 Jan 2014
Posts: 38

Kudos [?]: 16 [2], given: 5

Location: United States
GMAT 1: 720 Q47 V41
GPA: 3.71
Re: When the digits of two-digit, positive integer M are reversed, the res [#permalink]

Show Tags

New post 21 May 2016, 14:59
2
This post received
KUDOS
4
This post was
BOOKMARKED
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.

Kudos [?]: 16 [2], given: 5

Intern
Intern
avatar
Joined: 20 Sep 2016
Posts: 2

Kudos [?]: [0], given: 37

Location: Spain
GMAT 1: 740 Q50 V40
GPA: 2.45
Re: When the digits of two-digit, positive integer M are reversed, the res [#permalink]

Show Tags

New post 30 Dec 2016, 08:29
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!

Kudos [?]: [0], given: 37

Expert Post
1 KUDOS received
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 42269

Kudos [?]: 132841 [1], given: 12378

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

Show Tags

New post 30 Dec 2016, 08:34
1
This post received
KUDOS
Expert's post
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.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Kudos [?]: 132841 [1], given: 12378

Retired Moderator
User avatar
B
Joined: 05 Jul 2006
Posts: 1749

Kudos [?]: 443 [0], given: 49

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

Show Tags

New post 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

from 2

telling us what we already know ... insuff

A

Kudos [?]: 443 [0], given: 49

Director
Director
avatar
S
Joined: 12 Nov 2016
Posts: 794

Kudos [?]: 36 [0], given: 165

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

Show Tags

New post 15 Apr 2017, 16:35
thefibonacci 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.

Official Explanation:
[Reveal] Spoiler:
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

Kudos [?]: 36 [0], given: 165

Director
Director
avatar
S
Joined: 12 Nov 2016
Posts: 794

Kudos [?]: 36 [0], given: 165

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

Show Tags

New post 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.

The correct answer is A.

Kudos [?]: 36 [0], given: 165

Director
Director
avatar
S
Joined: 12 Nov 2016
Posts: 794

Kudos [?]: 36 [0], given: 165

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

Show Tags

New post 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?

Kudos [?]: 36 [0], given: 165

Intern
Intern
avatar
B
Joined: 28 Jun 2017
Posts: 6

Kudos [?]: 0 [0], given: 14

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

Show Tags

New post 07 Sep 2017, 05:05
AutoBot wrote:
I solved the question like this :
As per the give info , M = 10a + b ; N =10b + a
Also, M>N

Lets analyse statements now:
Stmt 1 : The integer (M - N) has 12 unique factors.

M - N = (10a + b) - (10b + a) = 9a - 9b
so, M - N = 9 (a-b) = 3^2 * (a-b)

Stmt1 says M - N has 12 factors , this implies that (a-b) should be a number with power as 3.

Remember the rule , if prime factorization of the integer ,N = X^p * Y^q * Z^r , then the number of factors of N = (p+1)*(q+1)*(r+1)

So, a-b should be a number with power as 3. This implies it should be 8 , which is 2^3.

Hence, (a - b) = 8 , this implies a= 9 and b =1

So, M= 91 and N = 19 , coz given is M>N

Stmt1: Sufficient

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

M - N = (10a + b) - (10b + a) = 9a - 9b
so, M - N = 9 (a-b) . This is already a multiple of 9.

Therefore, (a-b) can be any integer. Hence we cannot narrow down the values of a & b to find M.

Stmt2 : Insufficient



Hello, thanks for the explanation. But regarding statement 1, can't a-b be equal to -2 which would eventually make N>M. (therefore IS)

Can someone please explain?

Thanks!

Kudos [?]: 0 [0], given: 14

Re: When the digits of two-digit, positive integer M are reversed, the res   [#permalink] 07 Sep 2017, 05:05
Display posts from previous: Sort by

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

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  


GMAT Club MBA Forum Home| About| Terms and Conditions| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.