GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 22 Sep 2018, 00:27

### 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

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

# A positive integer x has 60 divisors and 7x has 80 divisors. What is t

Author Message
TAGS:

### Hide Tags

Senior Manager
Joined: 13 Jun 2013
Posts: 277
A positive integer x has 60 divisors and 7x has 80 divisors. What is t  [#permalink]

### Show Tags

22 Jan 2015, 02:01
3
9
00:00

Difficulty:

95% (hard)

Question Stats:

45% (01:58) correct 55% (01:40) wrong based on 121 sessions

### HideShow timer Statistics

A positive integer x has 60 divisors and 7x has 80 divisors. What is the greatest integer y such that $$7^y$$ divides n?

a) 0
b) 1
c) 2
d) 3
e) 4
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8284
Location: Pune, India
Re: A positive integer x has 60 divisors and 7x has 80 divisors. What is t  [#permalink]

### Show Tags

22 Jan 2015, 02:43
3
1
manpreetsingh86 wrote:
A positive integer x has 60 divisors and 7x has 80 divisors. What is the greatest integer y such that $$7^y$$ divides n?

a) 0
b) 1
c) 2
d) 3
e) 4

Total number of factors of x = 60 = (p+1)*(q+1)*(r+1)... = 2^2 * 3 * 5

Now note that 7x has only one 7 more than x. The number of all other prime factors stays the same.

Total number of factors of 7x = 80 = (p+2)*(q+1)*(r+1)... = 2^4 * 5

Here, the 3 of previous expression has disappeared so it must have converted to 4. Does it make sense? Let's see:
Total number of factors of x = 60 = 2^2 * 3 * 5 = (3+1)*(2+1)*(4+1)
Total number of factors of 7x = 80 = 2^2 * 4 * 5 = (3+1)*(3+1)*(4+1)

Perfect!

The powers of other factors stay the same. Only the power of 7 increases by 1. So initially, in x, the power of 7 must have been 2. Hence the maximum value of y must be 2.

_________________

Karishma
Veritas Prep GMAT Instructor

GMAT self-study has never been more personalized or more fun. Try ORION Free!

Current Student
Status: The Final Countdown
Joined: 07 Mar 2013
Posts: 285
Concentration: Technology, General Management
GMAT 1: 710 Q47 V41
GPA: 3.84
WE: Information Technology (Computer Software)
Re: A positive integer x has 60 divisors and 7x has 80 divisors. What is t  [#permalink]

### Show Tags

10 Feb 2015, 11:51
I am still unable to understand this explanation..
Total number of factors of x = 60 = 2^2 * 3 * 5(Its fine till here) = (3+1)*(2+1)*(4+1)(why this??)
Total number of factors of 7x = 80 = 2^2 * 4 * 5(Its fine till here) = (3+1)*(3+1)*(4+1)(why this??)

And this part leaves me

The powers of other factors stay the same. Only the power of 7 increases by 1. So initially, in x, the power of 7 must have been 2. Hence the maximum value of y must be 2.

Could you please elaborate on this?How can a 3 become a 4?and why?and how does 7 come into the picture?

TIA!
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8284
Location: Pune, India
Re: A positive integer x has 60 divisors and 7x has 80 divisors. What is t  [#permalink]

### Show Tags

10 Feb 2015, 22:38
1
Ralphcuisak wrote:
I am still unable to understand this explanation..
Total number of factors of x = 60 = 2^2 * 3 * 5(Its fine till here) = (3+1)*(2+1)*(4+1)(why this??)
Total number of factors of 7x = 80 = 2^2 * 4 * 5(Its fine till here) = (3+1)*(3+1)*(4+1)(why this??)

And this part leaves me

The powers of other factors stay the same. Only the power of 7 increases by 1. So initially, in x, the power of 7 must have been 2. Hence the maximum value of y must be 2.

Could you please elaborate on this?How can a 3 become a 4?and why?and how does 7 come into the picture?

TIA!

Tia, I think you need to check out this post first: http://www.veritasprep.com/blog/2010/12 ... ly-number/

This tells you that if N is of the form a^p*b^q*c^r..., the Total Number of Factors of a number N = (p+1)(q+1)(r+1)... (note that this is the number of factors of N, not N itself)

Here you are given that total number of factors on x is 60. Remember, x is not 60. The total number of factors of x is 60.
You need to write 60 in the form (p+1)(q+1)(r+1)...
One way of doing that is $$60 = 4*3*5 = (3+1)*(2+1)*(4+1)$$ Note that 4 = 3+1, 3 = 2+1, 5 = 4+1. So we have done nothing except changed the form.

When will we write that total number of factors of x are (3+1)*(2+1)*(4+1)? This happens when x is of the form $$a^3 * b^2 * c^4$$
_________________

Karishma
Veritas Prep GMAT Instructor

GMAT self-study has never been more personalized or more fun. Try ORION Free!

Math Expert
Joined: 02 Aug 2009
Posts: 6798
Re: A positive integer x has 60 divisors and 7x has 80 divisors. What is t  [#permalink]

### Show Tags

11 Feb 2015, 21:27
3
1
VeritasPrepKarishma wrote:
manpreetsingh86 wrote:
A positive integer x has 60 divisors and 7x has 80 divisors. What is the greatest integer y such that $$7^y$$ divides n?

a) 0
b) 1
c) 2
d) 3
e) 4

Total number of factors of x = 60 = (p+1)*(q+1)*(r+1)... = 2^2 * 3 * 5

Now note that 7x has only one 7 more than x. The number of all other prime factors stays the same.

Total number of factors of 7x = 80 = (p+2)*(q+1)*(r+1)... = 2^4 * 5

Here, the 3 of previous expression has disappeared so it must have converted to 4. Does it make sense? Let's see:
Total number of factors of x = 60 = 2^2 * 3 * 5 = (3+1)*(2+1)*(4+1)
Total number of factors of 7x = 80 = 2^2 * 4 * 5 = (3+1)*(3+1)*(4+1)

Perfect!

The powers of other factors stay the same. Only the power of 7 increases by 1. So initially, in x, the power of 7 must have been 2. Hence the maximum value of y must be 2.

hi,
you have given a good way of solving it .. kudos for that

i can think of one straight way to do it..
let the number(x..n..?) be a^k*b^l...7^t..
here we are just interested in value of 't'.
two points now..

1) n=a^k*b^l...7^t.. so number of factors=(k+1)(l+1)..(t+1)=60..
lets take all other values as z ie z=(k+1)(l+1).....
so z(t+1)=60....(1)

2)7n=a^k*b^l...7^(1+t).. so number of factors=(k+1)(l+1)..(t+2)=80..
as we take all other values as z ie z=(k+1)(l+1).....
so z(t+2)=80....(2)

from eq (1)and(2)...
80(t+1)=60(t+2)... t=2..
so ans is 2.. C
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

GMAT online Tutor

Intern
Joined: 07 Feb 2015
Posts: 3
Re: A positive integer x has 60 divisors and 7x has 80 divisors. What is t  [#permalink]

### Show Tags

11 Feb 2015, 21:42
chetan2u wrote:
VeritasPrepKarishma wrote:
manpreetsingh86 wrote:
A positive integer x has 60 divisors and 7x has 80 divisors. What is the greatest integer y such that $$7^y$$ divides n?

a) 0
b) 1
c) 2
d) 3
e) 4

Total number of factors of x = 60 = (p+1)*(q+1)*(r+1)... = 2^2 * 3 * 5

Now note that 7x has only one 7 more than x. The number of all other prime factors stays the same.

Total number of factors of 7x = 80 = (p+2)*(q+1)*(r+1)... = 2^4 * 5

Here, the 3 of previous expression has disappeared so it must have converted to 4. Does it make sense? Let's see:
Total number of factors of x = 60 = 2^2 * 3 * 5 = (3+1)*(2+1)*(4+1)
Total number of factors of 7x = 80 = 2^2 * 4 * 5 = (3+1)*(3+1)*(4+1)

Perfect!

The powers of other factors stay the same. Only the power of 7 increases by 1. So initially, in x, the power of 7 must have been 2. Hence the maximum value of y must be 2.

hi,
you have given a good way of solving it .. kudos for that

i can think of one straight way to do it..
let the number(x..n..?) be a^k*b^l...7^t..
here we are just interested in value of 't'.
two points now..

1) n=a^k*b^l...7^t.. so number of factors=(k+1)(l+1)..(t+1)=60..
lets take all other values as z ie z=(k+1)(l+1).....
so z(t+1)=60....(1)

2)7n=a^k*b^l...7^(1+t).. so number of factors=(k+1)(l+1)..(t+2)=80..
as we take all other values as z ie z=(k+1)(l+1).....
so z(t+2)=80....(2)

from eq (1)and(2)...
80(t+1)=60(t+2)... t=2..
so ans is 2.. C

hi chetan2u,
super bro.. how could simplify it so easily ..
Senior Manager
Status: Math is psycho-logical
Joined: 07 Apr 2014
Posts: 421
Location: Netherlands
GMAT Date: 02-11-2015
WE: Psychology and Counseling (Other)
Re: A positive integer x has 60 divisors and 7x has 80 divisors. What is t  [#permalink]

### Show Tags

02 Mar 2015, 14:10
VeritasPrepKarishma wrote:
manpreetsingh86 wrote:
A positive integer x has 60 divisors and 7x has 80 divisors. What is the greatest integer y such that $$7^y$$ divides n?

a) 0
b) 1
c) 2
d) 3
e) 4

Total number of factors of x = 60 = (p+1)*(q+1)*(r+1)... = 2^2 * 3 * 5

Now note that 7x has only one 7 more than x. The number of all other prime factors stays the same.

Total number of factors of 7x = 80 = (p+2)*(q+1)*(r+1)... = 2^4 * 5

Here, the 3 of previous expression has disappeared so it must have converted to 4. Does it make sense? Let's see:
Total number of factors of x = 60 = 2^2 * 3 * 5 = (3+1)*(2+1)*(4+1)
Total number of factors of 7x = 80 = 2^2 * 4 * 5 = (3+1)*(3+1)*(4+1)

Perfect!

The powers of other factors stay the same. Only the power of 7 increases by 1. So initially, in x, the power of 7 must have been 2. Hence the maximum value of y must be 2.

One question I have refers to the difference of x and 7x. What do we mean with "7x has only one 7 more than x" and with "Only the power of 7 increases by 1"?

I do understand the difference between (3+1)*(2+1)*(4+1) and (3+1)*(3+1)*(4+1), and I can see that the 2 became a 3. But I don't know how this relates to 7...
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8284
Location: Pune, India
Re: A positive integer x has 60 divisors and 7x has 80 divisors. What is t  [#permalink]

### Show Tags

02 Mar 2015, 20:04
1
pacifist85 wrote:
One question I have refers to the difference of x and 7x. What do we mean with "7x has only one 7 more than x" and with "Only the power of 7 increases by 1"?

I do understand the difference between (3+1)*(2+1)*(4+1) and (3+1)*(3+1)*(4+1), and I can see that the 2 became a 3. But I don't know how this relates to 7...

Ask yourself, why does 2 become 3?

Say $$x = 2^3 * 7^2 * 11^4$$
Total number of factors of $$x = (3+1)*(2+1)*(4+1) = 60$$

What will be 7x?

$$7x = 2^3 * 7^3 * 11^4$$
When you prime factorize 7x, you will get another 7.
Total number of factors of $$7x = (3+1)*(3+1)*(4+1) = 80$$
_________________

Karishma
Veritas Prep GMAT Instructor

GMAT self-study has never been more personalized or more fun. Try ORION Free!

Senior Manager
Status: Math is psycho-logical
Joined: 07 Apr 2014
Posts: 421
Location: Netherlands
GMAT Date: 02-11-2015
WE: Psychology and Counseling (Other)
Re: A positive integer x has 60 divisors and 7x has 80 divisors. What is t  [#permalink]

### Show Tags

03 Mar 2015, 01:52
VeritasPrepKarishma wrote:
pacifist85 wrote:
One question I have refers to the difference of x and 7x. What do we mean with "7x has only one 7 more than x" and with "Only the power of 7 increases by 1"?

I do understand the difference between (3+1)*(2+1)*(4+1) and (3+1)*(3+1)*(4+1), and I can see that the 2 became a 3. But I don't know how this relates to 7...

Ask yourself, why does 2 become 3?

Say $$x = 2^3 * 7^2 * 11^4$$
Total number of factors of $$x = (3+1)*(2+1)*(4+1) = 60$$

What will be 7x?

$$7x = 2^3 * 7^3 * 11^4$$
When you prime factorize 7x, you will get another 7.
Total number of factors of $$7x = (3+1)*(3+1)*(4+1) = 80$$

So, the point it that you can get whatever number using this number of factors for x: (3+1)*(2+1)*(4+1).
Then, you see than when x becomes 7x, the number of factors of one of the bases increases: (3+1)*(3+1)*(4+1).

Since the only thing that changed is that 7x has an additional prime factor (7), we conclude that this 7 belonged to (2+1) that became (3+1). Then we know that it was 2 before and this is how we get to our answer.
Manager
Joined: 14 Jul 2014
Posts: 182
Location: United States
Schools: Duke '20 (D)
GMAT 1: 600 Q48 V27
GMAT 2: 720 Q50 V37
GPA: 3.2
Re: A positive integer x has 60 divisors and 7x has 80 divisors. What is t  [#permalink]

### Show Tags

11 Apr 2016, 09:08
How is n related to x? I didn't understand this question at all..
Senior Manager
Joined: 03 Apr 2013
Posts: 283
Location: India
Concentration: Marketing, Finance
GMAT 1: 740 Q50 V41
GPA: 3
A positive integer x has 60 divisors and 7x has 80 divisors. What is t  [#permalink]

### Show Tags

31 Oct 2016, 04:53
manpreetsingh86 wrote:
A positive integer x has 60 divisors and 7x has 80 divisors. What is the greatest integer y such that $$7^y$$ divides n?

a) 0
b) 1
c) 2
d) 3
e) 4

Here's another method to do this. I have used algebraic method here..
First of all..notice that a power of 7 has to be a part of n already for the given condition to be true, because if that is not the case..then a completely new 7 will make the number of factors as 120(exactly double).
The number of factors of n can be written as
(a+1)(b+1)(c+1)...(s+1)(t+1) = 60...(t+1) is for powers of 7 ------ eq(1)

Let (a+1)(b+1)...(s+1) = k

According to the next condition..
(a+1)(b+1)....(s+1)(t+2) = 80

Or

k(t+2) = 80
=> k(t+1+1) = 80
=> k(t+1) + k = 80
Using the given value
=> 60 + k = 80
=> k = 20
Using this value in eq(1)

(t+1) = 3
=> t = 2

And Voila!

_________________

Spread some love..Like = +1 Kudos

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8284
Location: Pune, India
Re: A positive integer x has 60 divisors and 7x has 80 divisors. What is t  [#permalink]

### Show Tags

31 Oct 2016, 05:42
dina98 wrote:
How is n related to x? I didn't understand this question at all..

Note that n is a typo. The question is "What is the greatest integer y such that 7^y divides x?
_________________

Karishma
Veritas Prep GMAT Instructor

GMAT self-study has never been more personalized or more fun. Try ORION Free!

Senior Manager
Joined: 13 Oct 2016
Posts: 367
GPA: 3.98
A positive integer x has 60 divisors and 7x has 80 divisors. What is t  [#permalink]

### Show Tags

31 Oct 2016, 10:06
1
As I understand there should be x instead of n.
If x has 60 divisors we can represent it as follows:

1*60; 2*30; 3*20; 4*15 ; 5*12 ; 6*10 ; 2*2*15 ; 2*3*10 ; 2*2*3*5
$$p^{59}$$ ; $$p*q^{29}$$ ; $$p^2*q^{19}$$ ; $$p^3*q^{14}$$ ; $$p^4*q^{11}$$ ... (etc. - power of prime is 1 less than its representation in factor grouping)

Now if 7 is not present in initial number x, then in 7*x we’ll have 120 as total number of factors (we are adding 2 additional choices for additional prime and due to multiplication rule the whole result will be doubled).

So the only possibility is that 7 is already a prime factor of x. In this case, we just need to increase its power by 1. The obvious choice we can spot at first glance is 3*20. In order to get 80 we need 4*20. So the initial power of 7 in x was one less than 3.

Kind of amateurish by I hope understandable
Senior Manager
Joined: 02 Apr 2014
Posts: 477
GMAT 1: 700 Q50 V34
A positive integer x has 60 divisors and 7x has 80 divisors. What is t  [#permalink]

### Show Tags

01 Jan 2018, 21:15
I solved it like this:

First thing to note: if x does not have 7 as one of the prime factors, then 7x will have double the factors i.e. 120

so x already has 7 as one of the prime factors.
say prime factorization of $$x = (p + 1)(q+1)(r+1).... = 60$$ (total factors)
Let p be the power of 7 in x

then prime factorization of $$7x = (p+1+1)(q+1)(r+1).... = 80$$
substituting $$(q+1)(r+1)....$$ as = $$60/(p+1)$$
=> $$(p+2) * 60/(p+1) = 80$$
=> $$3p + 6 = 4p + 4$$ => $$p = 2$$
so x has $$7^2$$ has the factor.

So maximum y such that $$7^y$$ will divide x is 2 => (C)
A positive integer x has 60 divisors and 7x has 80 divisors. What is t &nbs [#permalink] 01 Jan 2018, 21:15
Display posts from previous: Sort by

# Events & Promotions

 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®.