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

 It is currently 15 Dec 2019, 06:49 ### 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

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both posit

Author Message
TAGS:

### Hide Tags

Manager  Joined: 01 Sep 2012
Posts: 114
If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both posit  [#permalink]

### Show Tags

5
26 00:00

Difficulty:   55% (hard)

Question Stats: 56% (01:31) correct 44% (01:41) wrong based on 468 sessions

### HideShow timer Statistics

If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both positive integers, how many different possible values of n are there?

A. 1
B. 2
C. 3
D. 4
E. 6

So what I did is basically making 35^n = (5^n)(7^n)

Thought the answer is B but i was wrong.
Can anyone explain?
Magoosh GMAT Instructor G
Joined: 28 Dec 2011
Posts: 4473
Re: If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both posit  [#permalink]

### Show Tags

4
2
roygush wrote:
If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both positive integers, how many different possible values of n are there?

A.1
B.2
C.3
D.4
E.6

So what I did is basically making 35^n = (5^n)(7^n)

Thought the answer is B but i was wrong.
Can anyone explain?

Dear Roygush

I must say, starting from 35^n = (5^n)(7^n), it's not clear to me how you wound up with only two possibilities. Your reasoning is not clear to me. I would say --- start on the left side ---

The factor of (3^4) doesn't play into the (35^n) at all --- that will have to go into the x, no choice.

We have three factors of 7, so that means we could have as many as three powers of 35 ------ we could have 35^1, 35^2, or 35^3. We can't have 35^0 = 1, because even though 1 is positive integer, that would make n = 0, and zero is not a positive integer. So, those three values of n, 1 and 2 and 3, are the only possibilities. Therefore, there are three possibilities.

n = 1 -------> x = (3^4)(5^5)(7^2)
n = 2 -------> x = (3^4)(5^4)(7^1)
n = 3 -------> x = (3^4)(5^3)

Does all that make sense?

Mike _________________
Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
##### General Discussion
Intern  Joined: 08 Dec 2012
Posts: 7
Re: If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both posit  [#permalink]

### Show Tags

2
(3^4) (5^3) (5^3) (7^3) = (3^4) (5^3) (35^3) => n = 3, x = (3^4) (5^3)
Math Expert V
Joined: 02 Sep 2009
Posts: 59728
Re: If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both posit  [#permalink]

### Show Tags

4
2
If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both positive integers, how many different possible values of n are there?

A. 1
B. 2
C. 3
D. 4
E. 6

Notice that the power of 7 in LHS is limiting the value of n, thus n cannot be more than 3 and since n is a positive integer, then n could be 1, 2, or 3.

_________________
Manager  Joined: 01 Sep 2012
Posts: 114
Re: If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both posit  [#permalink]

### Show Tags

thank you all.
Its clear now and i should have thought about it myself.
Magoosh GMAT Instructor G
Joined: 28 Dec 2011
Posts: 4473
Re: If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both posit  [#permalink]

### Show Tags

1
This question turned out to be relatively easy, but it inspired me to create a similar question that's a little more challenging:
if-a-and-b-are-positive-integers-and-147953.html

Mike _________________
Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
Manager  Joined: 04 Mar 2013
Posts: 64
Location: India
Concentration: General Management, Marketing
GPA: 3.49
WE: Web Development (Computer Software)
Re: If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both posit  [#permalink]

### Show Tags

Bunuel wrote:
If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both positive integers, how many different possible values of n are there?

A. 1
B. 2
C. 3
D. 4
E. 6

Notice that the power of 7 in LHS is limiting the value of n, thus n cannot be more than 3 and since n is a positive integer, then n could be 1, 2, or 3.

hi, i differ on this,

here is the reason

35^0 is also possible as we are not given any limitation to x

we can have values of 35^0, 1, 2, 3, so n can take 4 values .......... and the answer would be 4 D
Math Expert V
Joined: 02 Sep 2009
Posts: 59728
Re: If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both posit  [#permalink]

### Show Tags

krrish wrote:
Bunuel wrote:
If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both positive integers, how many different possible values of n are there?

A. 1
B. 2
C. 3
D. 4
E. 6

Notice that the power of 7 in LHS is limiting the value of n, thus n cannot be more than 3 and since n is a positive integer, then n could be 1, 2, or 3.

hi, i differ on this,

here is the reason

35^0 is also possible as we are not given any limitation to x

we can have values of 35^0, 1, 2, 3, so n can take 4 values .......... and the answer would be 4 D

Please read the stem carefully: "x and n are both positive integers..." 0 is NOT a positive integer.
_________________
SVP  Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1727
Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)
Re: If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both posit  [#permalink]

### Show Tags

2
Re-arrange the equation:

x = 3^4 . 5^6. 7^3 / 35^n

n may be 0,1,2 3........ however n is +ve, so can be 1,2,3

Manager  Joined: 20 Dec 2013
Posts: 222
Location: India
Re: If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both posit  [#permalink]

### Show Tags

Option C.
RHS:7^n * 5^n * x
On the LHS,7 has max. Power of 3
Since n is a +ve integer,it can't be zero.So n=1,2,&3 only.

Posted from my mobile device
Intern  Joined: 22 Feb 2014
Posts: 2
Re: If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both posit  [#permalink]

### Show Tags

How is this problem solved? I don't understand why the powers of 7 limits the answer choices. Maybe I am thinking about this the wrong way.
Magoosh GMAT Instructor G
Joined: 28 Dec 2011
Posts: 4473
Re: If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both posit  [#permalink]

### Show Tags

2
frenchwr wrote:
How is this problem solved? I don't understand why the powers of 7 limits the answer choices. Maybe I am thinking about this the wrong way.

Dear frenchwr,
I'm happy to respond. How well do you understand the idea of the prime factorization of a number? See:
http://magoosh.com/gmat/2012/gmat-math-factors/

When you know the prime factorization of a number, it's as if you know the DNA of the number. You have such profound knowledge about it.

For example, 35 = 5*7. That's the prime factorization. This means, for every factor of 35 we have, we need another factor of 7. There is absolutely no way we can make a factor of 35 without using a factor of 7 as one of the ingredients. If we only get three factors of 7, as is evident from the left side of the equation, that means we could make, at most, only three factors of 35. Once those three factors of 7 are used up, building the three factors of 35, then there is absolutely no possible of creating any more factors of 35, because we are out of one of the essential ingredients.

Does this make sense?
Mike _________________
Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
CEO  D
Status: GMATINSIGHT Tutor
Joined: 08 Jul 2010
Posts: 2977
Location: India
GMAT: INSIGHT
Schools: Darden '21
WE: Education (Education)
Re: If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both posit  [#permalink]

### Show Tags

2
roygush wrote:
If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both positive integers, how many different possible values of n are there?

A. 1
B. 2
C. 3
D. 4
E. 6

So what I did is basically making 35^n = (5^n)(7^n)

Thought the answer is B but i was wrong.
Can anyone explain?

TIP FOR SUCH QUESTIONS: All such Questions Require PRIME FACTORISATION i.e. Breaking the number into Prime factors and their power form on both sides of the equation

(3^4)(5^6)(7^3) = (35^n)(x)

i.e. (3^4)(5^6)(7^3) = (5*7)^n *(x)

i.e. (3^4)(5^6)(7^3) = (5^n)*(7^n) *(x)

This clearly depicts one thing which is x must be a multiple of (3^4) for sure and other factors of x will depend on the value of n

Please not that n is a positive integer

i.e. at n=1, (3^4)(5^6)(7^3) = (5^1)*(7^1) *(x) i.e. x=(3^4)(5^5)(7^2)
i.e. at n=2, (3^4)(5^6)(7^3) = (5^2)*(7^2) *(x) i.e. x=(3^4)(5^4)(7^1)
i.e. at n=3, (3^4)(5^6)(7^3) = (5^3)*(7^3) *(x) i.e. x=(3^4)(5^3)(7^0)

Only 3 possibilities

_________________
Prosper!!!
GMATinsight
Bhoopendra Singh and Dr.Sushma Jha
e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772
Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi
http://www.GMATinsight.com/testimonials.html

ACCESS FREE GMAT TESTS HERE:22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION
Current Student D
Joined: 12 Aug 2015
Posts: 2552
Schools: Boston U '20 (M)
GRE 1: Q169 V154 Re: If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both posit  [#permalink]

### Show Tags

roygush wrote:
If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both positive integers, how many different possible values of n are there?

A. 1
B. 2
C. 3
D. 4
E. 6

So what I did is basically making 35^n = (5^n)(7^n)

Thought the answer is B but i was wrong.
Can anyone explain?

here => 3^4 is immaterial
now values of n can be found from the following expressions
on LHS there can be (35)^1 or (35)^2 or (35)^3 but beyond that we dont have the excess power of 5 to utilize
thus => N=1,2,3 => 3 values.
_________________
Magoosh GMAT Instructor G
Joined: 28 Dec 2011
Posts: 4473
Re: If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both posit  [#permalink]

### Show Tags

1
Chiragjordan wrote:
roygush wrote:
If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both positive integers, how many different possible values of n are there?

A. 1
B. 2
C. 3
D. 4
E. 6

So what I did is basically making 35^n = (5^n)(7^n)

Thought the answer is B but i was wrong.
Can anyone explain?

here => 3^4 is immaterial
now values of n can be found from the following expressions
on LHS there can be (35)^1 or (35)^2 or (35)^3 but beyond that we dont have the excess power of 5 to utilize
thus => N=1,2,3 => 3 values.

Dear Chiragjordan,
Yes, you are quite right. The 3^4 is entirely immaterial, and the factors of 7 limit the possibilities. This is a relatively straightforward problem. This problem inspired me to create a slightly more challenging problem:
if-a-and-b-are-positive-integers-and-147953.html
Enjoy!
Mike _________________
Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
VP  P
Joined: 12 Dec 2016
Posts: 1485
Location: United States
GMAT 1: 700 Q49 V33 GPA: 3.64
Re: If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both posit  [#permalink]

### Show Tags

oh man, i miss the word positive, so i thought n can be o, so i chose D
Intern  B
Joined: 09 Mar 2017
Posts: 34
Location: India
GMAT 1: 650 Q45 V31 GPA: 4
Re: If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both posit  [#permalink]

### Show Tags

Hi !
If in the case the value of n =4 ; we still get a value which +ve decimal..
Why cant we say then it has more values
Math Expert V
Joined: 02 Sep 2009
Posts: 59728
Re: If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both posit  [#permalink]

### Show Tags

amitpandey25 wrote:
Hi !
If in the case the value of n =4 ; we still get a value which +ve decimal..
Why cant we say then it has more values

If n = 4, then from (3^4)(5^6)(7^3) = (35^4)(x), x turns out to be x = 2025/7 but the stem says that x is a positive integer, thus n cannot be 4.
_________________
Intern  B
Joined: 09 Mar 2017
Posts: 34
Location: India
GMAT 1: 650 Q45 V31 GPA: 4
Re: If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both posit  [#permalink]

### Show Tags

Oh ! Thanks Bunuel Non-Human User Joined: 09 Sep 2013
Posts: 13743
Re: If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both posit  [#permalink]

### Show Tags

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________ Re: If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both posit   [#permalink] 01 Jul 2019, 08:04
Display posts from previous: Sort by

# If (3^4)(5^6)(7^3) = (35^n)(x), where x and n are both posit   