It is currently 24 Feb 2018, 00:09

### 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

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# If x is an integer, is 3^x a factor of 15! ?

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 43894
If x is an integer, is 3^x a factor of 15! ? [#permalink]

### Show Tags

25 Aug 2015, 23:42
Expert's post
5
This post was
BOOKMARKED
00:00

Difficulty:

45% (medium)

Question Stats:

70% (01:10) correct 30% (01:04) wrong based on 181 sessions

### HideShow timer Statistics

If x is an integer, is 3^x a factor of 15! ?

(1) x is the sum of two distinct single-digit prime numbers.
(2) 0 < x < 6

Kudos for a correct solution.
[Reveal] Spoiler: OA

_________________
Manager
Joined: 13 Mar 2013
Posts: 176
Location: United States
GPA: 3.5
WE: Engineering (Telecommunications)
Re: If x is an integer, is 3^x a factor of 15! ? [#permalink]

### Show Tags

26 Aug 2015, 01:32
1
KUDOS
15! / 3^x remainder is zero .

that means the max. value of x can be 6 and min.value be 0 .

1)x= sum of two prime number ( can be 5 ) then yes
x= 2+7 = 9 then no

So one not sufficient

2) 0<x<6 all the value of of x satisfy the main equation .

B ok and hence ans.
_________________

Regards ,

Current Student
Joined: 20 Mar 2014
Posts: 2683
Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44
GPA: 3.7
WE: Engineering (Aerospace and Defense)
Re: If x is an integer, is 3^x a factor of 15! ? [#permalink]

### Show Tags

26 Aug 2015, 03:15
2
KUDOS
1
This post was
BOOKMARKED
Bunuel wrote:
If x is an integer, is 3^x a factor of 15! ?

(1) x is the sum of two distinct single-digit prime numbers.
(2) 0 < x < 6

Kudos for a correct solution.

x $$\in$$integer.

Is 15! = 3^x * p ?

Greatest power of 3 in 15! = 15/3 + 15/$$3^2$$ = 5+1 = 6. Thus if 0< x < 6 , then we would have a "yes" for the question asked.

Per statement 1, you can have 2+3 =5 "yes" but you can also have 3+5 = 8 " no". Thus not sufficient.

Per statement 2, 0<x<6 ---> Exactly what we wanted. Sufficient.

Manager
Joined: 14 Mar 2014
Posts: 148
GMAT 1: 710 Q50 V34
Re: If x is an integer, is 3^x a factor of 15! ? [#permalink]

### Show Tags

26 Aug 2015, 09:29
Bunuel wrote:
If x is an integer, is 3^x a factor of 15! ?

(1) x is the sum of two distinct single-digit prime numbers.
(2) 0 < x < 6

Kudos for a correct solution.

IMO : B

Highest power of 3 in 15! = 6

St 1 : x is the sum of two distinct single-digit prime numbers.

Prime = {2,3,5,7}
for sum(2,3) = 2+3 = 5 ... Satisfies
for sum(5,7) = 5+7 = 12 ... Doesn't satisfy
Hence not suff

St 2 : 0 < x < 6
Since highest power of 3 in 15! can be 6.
Max value of x = 6
Hence given condition is well in the range.

Hence suff
_________________

I'm happy, if I make math for you slightly clearer
And yes, I like kudos
¯\_(ツ)_/¯

Manager
Joined: 08 Sep 2012
Posts: 66
Location: India
Concentration: Social Entrepreneurship, General Management
WE: Engineering (Investment Banking)
Re: If x is an integer, is 3^x a factor of 15! ? [#permalink]

### Show Tags

26 Aug 2015, 10:25
1
KUDOS
Bunuel wrote:
If x is an integer, is 3^x a factor of 15! ?

(1) x is the sum of two distinct single-digit prime numbers.
(2) 0 < x < 6

Kudos for a correct solution.

For 3^x to be a factor of 15! => 15! should be divisible by 3^x, leaving no remainder

Power of 3 in 15! = 15/3 + 15/9 = 5 + 1 = 6 => x <= 6.
Note that x can be negative too, and for all negative integer values of x, 15! / 3^x will result in an integer. The only constraint is that x <= 6

St. 1 => X is sum of two distinct single-digit prime numbers4
=> x = 2+3 = 5 => Answer is Yes & x = 2+5=7 => Answer is No => Not sufficient

St. 2 => 0 < x < 6 => Answer is always Yes as discussed above => Sufficient

_________________

+1 Kudos if you liked my post! Thank you!

Manager
Joined: 14 Jul 2014
Posts: 191
Location: United States
GMAT 1: 600 Q48 V27
GMAT 2: 720 Q50 V37
GPA: 3.2
Re: If x is an integer, is 3^x a factor of 15! ? [#permalink]

### Show Tags

26 Aug 2015, 12:24
Should not it be D? The first statement may work even if X is 5+7. If X = 12, 36 can be a factor of 15!. Am I wrong?
15! = 15*.... * 12 * .. 3 * 2 * 1.
Current Student
Joined: 20 Mar 2014
Posts: 2683
Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44
GPA: 3.7
WE: Engineering (Aerospace and Defense)
If x is an integer, is 3^x a factor of 15! ? [#permalink]

### Show Tags

26 Aug 2015, 12:48
1
This post was
BOOKMARKED
dina98 wrote:
Should not it be D? The first statement may work even if X is 5+7. If X = 12, 36 can be a factor of 15!. Am I wrong?
15! = 15*.... * 12 * .. 3 * 2 * 1.

No. Statement 1 is not sufficient. Look below.

if x = 12, then $$3^{12}$$should be a factor of 15!

For maximum power of $$A^n$$in M! , we use the formula: M/A + M/$$A^2$$ + M/$$A^3$$ .... M/$$A^n$$ where $$A^n$$< M

Thus, maximum p9ower of 3 in 15! = 15/3+ 15/3^2 = 5+1 = 6. So in other words, 3,$$3^2$$,$$3^3$$, $$3^4$$, $$3^5$$, $$3^6$$ are the only powers of 3 that will give you as a factor of 15!. Anything greater than $$3^6$$ ($$3^7$$ or $$3^{12}$$) will not give you an integer when you calculate 15!/$$3^{12}$$.

I believe you are confusing '^' symbol with the symbol '*'.

"^" is 'raised to the power of' and NOT the multiplication symbol. We write multiplication symbol as either 'x' or '*'

3^12 = $$3^{12}$$= 3*3*3*3*3*3*3*3*3*3*3*3 $$\neq$$ 36

3*12 = 36

Hope this helps.
Manager
Joined: 14 Jul 2014
Posts: 191
Location: United States
GMAT 1: 600 Q48 V27
GMAT 2: 720 Q50 V37
GPA: 3.2
If x is an integer, is 3^x a factor of 15! ? [#permalink]

### Show Tags

26 Aug 2015, 12:52
Engr2012 wrote:

I believe you are confusing '^' symbol with the symbol '*'.

"^" is 'raised to the power of' and NOT the multiplication symbol. We write multiplication symbol as either 'x' or '*'

3^12 = 3*3*3*3*3*3*3*3*3*3*3*3 $$\neq$$36

3*12 = 36

Hope this helps.

Ahh yes, that makes sense. Confused the symbols. Thank you!
Senior Manager
Joined: 28 Feb 2014
Posts: 296
Location: United States
Concentration: Strategy, General Management
Re: If x is an integer, is 3^x a factor of 15! ? [#permalink]

### Show Tags

26 Aug 2015, 17:56
1
KUDOS
If x is an integer, is 3^x a factor of 15! ?
reworded,
is x<=6?
this is because 15! has six 3's aka 3^6

(1) x is the sum of two distinct single-digit prime numbers.
x=2+3=5 yes
x=3+5=8 no
Insufficient

(2) 0 < x < 6
x is less than 6
Sufficient

Senior Manager
Joined: 09 Feb 2015
Posts: 388
Location: India
Concentration: Social Entrepreneurship, General Management
GMAT 1: 690 Q49 V34
GMAT 2: 720 Q49 V39
GPA: 2.8
Re: If x is an integer, is 3^x a factor of 15! ? [#permalink]

### Show Tags

28 Aug 2015, 11:19
1
KUDOS
15! Has 6 3s so any number from 3^0 to 3^6 will be a factor of 15!..hence b

Sent from my iPhone using Tapatalk
Director
Joined: 21 May 2013
Posts: 581
Re: If x is an integer, is 3^x a factor of 15! ? [#permalink]

### Show Tags

29 Aug 2015, 05:12
1
KUDOS
Bunuel wrote:
If x is an integer, is 3^x a factor of 15! ?

(1) x is the sum of two distinct single-digit prime numbers.
(2) 0 < x < 6

Kudos for a correct solution.

For this we need to find the highest power of 3 in 15!

(15/3)+(15/9)=5+1=6
(1) x is the sum of two distinct single-digit prime numbers. 2,3,5,7 are the four possible prime nos. 2+3=5 looks good but 3+7=10 exceeds 6.Insufficient
(2) 0 < x < 6. Now, if xis more than 0 but less than 6, this is what we want since highest power of 3 in 15! is 6. Sufficient.

Current Student
Joined: 04 May 2015
Posts: 73
Concentration: Strategy, Operations
WE: Operations (Military & Defense)
If x is an integer, is 3^x a factor of 15! ? [#permalink]

### Show Tags

29 Aug 2015, 10:43
2
KUDOS
Bunuel wrote:
If x is an integer, is 3^x a factor of 15! ?

(1) x is the sum of two distinct single-digit prime numbers.
(2) 0 < x < 6

Kudos for a correct solution.

As per my normal approach for solving problems, where possible I try to avoid actual calculations in favour of more intuitive methods (don't have a strong math background).

If x is an integer, is 3^x a factor of 15! ? (this is a Yes or No question)

How many times does the number 3 turn up in 15!
15! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15
# of 3's =....1...........1...........2..............1...............1 = 6 x 3's

From this we can determine that as long as x <= 6 it will be divisible by 15! ----> if x <= 6 Answer will be YES (Sufficient), if x > 6 answer will be NO (Sufficient)

On to the statements...

(1) x is the sum of two distinct single-digit prime numbers... single digit primes are 2, 3, 5 & 7 ----> x is between 2 + 3 = 5 and 5 + 7 = 12 ----> INSUFFICIENT
(2) 0 < x < 6.... Based of the previous information we can definitively answer this question as YES ----> SUFFICIENT

P.S. as I have posted on a few questions now trying to build my own confidence (teaching helps me understand) but if you have any feedback on the way I am presenting it please let me know. Thanks in advance
_________________

If you found my post useful, please consider throwing me a Kudos... Every bit helps

Math Expert
Joined: 02 Sep 2009
Posts: 43894
Re: If x is an integer, is 3^x a factor of 15! ? [#permalink]

### Show Tags

30 Aug 2015, 08:05
1
KUDOS
Expert's post
Bunuel wrote:
If x is an integer, is 3^x a factor of 15! ?

(1) x is the sum of two distinct single-digit prime numbers.
(2) 0 < x < 6

Kudos for a correct solution.

PRINCETON REVIEW OFFICIAL SOLUTION:

This is a much more complicated question than the first one, and we really have to do some thinking up front before we worry about the statements. First, this is a Yes/No question, so we need to be clear that it doesn’t matter whether the answer is yes or no, whether 3^x is a factor of 15! or not. All that matters is that we know for certain one way or the other. So how will we know?

3^x describes a certain number of 3s multiplied together. If 3^x is a factor of 15! then all those 3s divide evenly into 15! with nothing left over (if z is a factor of some number then that number is divisible by z). The question becomes, “How many factors of 3 are there in 15! ?” Write out 15! and count the factors of 3.

15! = 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1.

There’s one in the 15 (3 x 5), one in the 12 (3 x 4), two in the 9 (3 x 3), one in the 6 (3 x 2) and one in the 3 (3 x 1). That’s a total of six 3s. So now we can finally say what the missing piece of the puzzle is. Since 15! contains six factors of 3, if x = 6 or anything less than 6 then 3^x will be a factor of 15! and the answer will be Yes. If x is anything larger than 6 then 3^x will not be a factor of 15! because it will have too many 3s. The answer will be No. So the key question we’re concerned with as we turn to the statements is do we know whether x<=6 or x>6 so we have answered our question definitively. Statement 2 is sufficient and the answer to the question is (B).
_________________
Non-Human User
Joined: 09 Sep 2013
Posts: 13776
Re: If x is an integer, is 3^x a factor of 15! ? [#permalink]

### Show Tags

29 Jan 2018, 23:01
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 x is an integer, is 3^x a factor of 15! ?   [#permalink] 29 Jan 2018, 23:01
Display posts from previous: Sort by