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# If p is the product of the integers from 1 to 30, inclusive

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If p is the product of the integers from 1 to 30, inclusive [#permalink]

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20 Jun 2010, 22:59
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If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?

A. 10
B. 12
C. 14
D. 16
E. 18
[Reveal] Spoiler: OA

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21 Jun 2010, 01:23
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divakarbio7 wrote:
If p is the product of the integers from 1 to 30,
inclusive, what is the greatest integer k for which 3k is
a factor of p ?
(A) 10
(B) 12
(C) 14
(D) 16
(E) 18

Not sure as to how to approach this problem

Given: $$p=30!=3^k*n$$. Question: $$k_{max}=?$$.

We should find highest power of 3 in 30!. Finding the power of a prime in n!: everything-about-factorials-on-the-gmat-85592.html or math-number-theory-88376.html (factorials chapter).

So:

$$\frac{30}{3}+\frac{30}{9}+\frac{30}{27}=10+3+1=14$$.

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23 Jun 2010, 12:38
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15. No need to actually find the product of 1 to 30. Just look at the numbers that have factors of 3.

3 = 3
6 = 3 * 2
9 = 3 * 3
12 = 3 * 2 * 2
15 = 3 * 5
18 = 3 * 3 * 2
21 = 3 * 7
24 = 3 * 2 * 2 * 2
27 = 3 * 3 * 3
30 = 3 * 2 * 5

There are 14 3's, so the largest factor of p for 3k is 3*14.

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26 Feb 2012, 10:19
$$\frac{30}{3}+\frac{30}{3^2}+\frac{30}{3^3}$$
= 10 + 3 + 1 (ignore the decimals part of each fraction)
= 14
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Re: official guide 13th ed #116 problem solving [#permalink]

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19 Jun 2012, 05:13
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peter2000 wrote:
p is product of integers from 1 to 30 inclusive, what is the greatest integer k for which3^k is a factor of p?

A)10
B)12
C)14
D)16
E)18

I understand the way the GMAT guide does the problem, I was wondering if there was a quicker way?

Check out this post for an explanation of the method you can use to solve such questions:
http://www.veritasprep.com/blog/2011/06 ... actorials/
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for \$199

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Kudos [?]: 18112 [5], given: 236

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Re: If p is the product of the integers from 1 to 30, inclusive [#permalink]

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01 Oct 2012, 05:50
P = 30!
8
P = 30 x 29 x 28 x 27 x 26 x 25 x 24 x 24 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13 x 12 x 11 x 10 x 09 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

Out of these 30 , 27 , 24 , 21 , 18, 15 , 12 , 09 , 06 , 3 are factors of 3

3 x 10 , 3 x 3 x 3 , 3 x 8 , 3 x 3 x 2, 3 x 5 , 3 x 4 , 3 x 3 x 3 , 3 x 2 , 3

So we have a total of 14 three's ...

Therefore the maximum value of K can be 14 (C)
_________________

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Re: If p is the product of the integers from 1 to 30, inclusive [#permalink]

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16 Aug 2016, 14:56
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Top Contributor
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divakarbio7 wrote:
If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?

A. 10
B. 12
C. 14
D. 16
E. 18

This question is really asking us to determine the number of 3's "hiding" in the prime factorization of p.

p = (1)(2)(3)(4)(5)(6)(7)(8)(9) . . . (27)(28)(29)(30)
= (1)(2)(3)(4)(5)(2)(3)(7)(8)(3)(3)(10)(11)(3)(4)(13)(14)(3)(5)(16)(17)(3)(3)(2)(19)(20)(3)(7)(22)(23)(3)(8)(25)(26). . . (3)(3)(3)(28)(29)(3)(10)
= (3)^14(other non-3 stuff)

[Reveal] Spoiler:
C

RELATED VIDEO

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Re: If p is the product of the integers from 1 to 30, inclusive [#permalink]

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24 Jan 2017, 18:52
1) We need to calculate the number of prime factors 3 in the product of all the integers from 1 to 30
2) 30/3=10; 30/9=3; 30/27=1; 10+3+1=14

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Re: If p is the product of the integers from 1 to 30, inclusive [#permalink]

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25 Mar 2017, 09:20
30/3, 30/9, 30/27
10+3+1=14

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Re: If p is the product of the integers from 1 to 30, inclusive [#permalink]

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25 Mar 2017, 23:35
If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?

A. 10
B. 12
C. 14
D. 16
E. 18

product of 1 to 30 includes the multiples of 3 and the number of 3 appear ,count them and add

3-1
6-1
9-2
12-1
15-1
18-2
21-1
24-1
27-3
30-1
=14

Kudos [?]: 81 [0], given: 18

Re: If p is the product of the integers from 1 to 30, inclusive   [#permalink] 25 Mar 2017, 23:35
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Given the text, the content structure is as follows:

The passage starts with a timestamp and a reference to a specific date, followed by a section header indicating "Decision(s) Day!" with links to chat rooms for various business schools. It then transitions into a section about GMAT Club Daily Prep, explaining the service and its features. The main content of the passage is a math problem related to factorials and prime factors.
single_question_with_answer

the passage contains a single mathematical question: "if p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?" the answer to this question is provided in the discussion, with the correct answer being 14.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
It is currently 13 Dec 2017, 10:58

# Decision(s) Day!:

CHAT Rooms | Ross R1 | Kellogg R1 | Darden R1 | Tepper R1

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# If p is the product of the integers from 1 to 30, inclusive

Author Message
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Intern
Joined: 29 Nov 2009
Posts: 25

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Location: Bangalore - INDIA
Schools: Duke, NUS,Rutgers
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If p is the product of the integers from 1 to 30, inclusive [#permalink]

### Show Tags

20 Jun 2010, 22:59
3
KUDOS
26
This post was
BOOKMARKED
00:00

Difficulty:

25% (medium)

Question Stats:

72% (00:59) correct 28% (01:26) wrong based on 1371 sessions

### HideShow timer Statistics

If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?

A. 10
B. 12
C. 14
D. 16
E. 18
[Reveal] Spoiler: OA

Kudos [?]: 70 [3], given: 3

Math Expert
Joined: 02 Sep 2009
Posts: 42583

Kudos [?]: 135518 [9], given: 12697

### Show Tags

21 Jun 2010, 01:23
9
KUDOS
Expert's post
16
This post was
BOOKMARKED
divakarbio7 wrote:
If p is the product of the integers from 1 to 30,
inclusive, what is the greatest integer k for which 3k is
a factor of p ?
(A) 10
(B) 12
(C) 14
(D) 16
(E) 18

Not sure as to how to approach this problem

Given: $$p=30!=3^k*n$$. Question: $$k_{max}=?$$.

We should find highest power of 3 in 30!. Finding the power of a prime in n!: everything-about-factorials-on-the-gmat-85592.html or math-number-theory-88376.html (factorials chapter).

So:

$$\frac{30}{3}+\frac{30}{9}+\frac{30}{27}=10+3+1=14$$.

_________________

Kudos [?]: 135518 [9], given: 12697

Manager
Joined: 30 May 2010
Posts: 189

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23 Jun 2010, 12:38
1
KUDOS
15. No need to actually find the product of 1 to 30. Just look at the numbers that have factors of 3.

3 = 3
6 = 3 * 2
9 = 3 * 3
12 = 3 * 2 * 2
15 = 3 * 5
18 = 3 * 3 * 2
21 = 3 * 7
24 = 3 * 2 * 2 * 2
27 = 3 * 3 * 3
30 = 3 * 2 * 5

There are 14 3's, so the largest factor of p for 3k is 3*14.

Kudos [?]: 261 [1], given: 32

Senior Manager
Status: mba here i come!
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26 Feb 2012, 10:19
$$\frac{30}{3}+\frac{30}{3^2}+\frac{30}{3^3}$$
= 10 + 3 + 1 (ignore the decimals part of each fraction)
= 14
_________________

press +1 Kudos to appreciate posts

Kudos [?]: 1261 [0], given: 48

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7792

Kudos [?]: 18112 [5], given: 236

Location: Pune, India
Re: official guide 13th ed #116 problem solving [#permalink]

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19 Jun 2012, 05:13
5
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Expert's post
2
This post was
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peter2000 wrote:
p is product of integers from 1 to 30 inclusive, what is the greatest integer k for which3^k is a factor of p?

A)10
B)12
C)14
D)16
E)18

I understand the way the GMAT guide does the problem, I was wondering if there was a quicker way?

Check out this post for an explanation of the method you can use to solve such questions:
http://www.veritasprep.com/blog/2011/06 ... actorials/
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for \$199

Veritas Prep Reviews

Kudos [?]: 18112 [5], given: 236

Senior Manager
Joined: 03 Sep 2012
Posts: 363

Kudos [?]: 257 [0], given: 37

Location: United States
Concentration: Healthcare, Strategy
GMAT 1: 730 Q48 V42
GPA: 3.88
WE: Medicine and Health (Health Care)
Re: If p is the product of the integers from 1 to 30, inclusive [#permalink]

### Show Tags

01 Oct 2012, 05:50
P = 30!
8
P = 30 x 29 x 28 x 27 x 26 x 25 x 24 x 24 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13 x 12 x 11 x 10 x 09 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

Out of these 30 , 27 , 24 , 21 , 18, 15 , 12 , 09 , 06 , 3 are factors of 3

3 x 10 , 3 x 3 x 3 , 3 x 8 , 3 x 3 x 2, 3 x 5 , 3 x 4 , 3 x 3 x 3 , 3 x 2 , 3

So we have a total of 14 three's ...

Therefore the maximum value of K can be 14 (C)
_________________

"When you want to succeed as bad as you want to breathe, then you’ll be successful.” - Eric Thomas

Kudos [?]: 257 [0], given: 37

SVP
Joined: 11 Sep 2015
Posts: 1906

Kudos [?]: 2743 [0], given: 364

Re: If p is the product of the integers from 1 to 30, inclusive [#permalink]

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16 Aug 2016, 14:56
Expert's post
Top Contributor
1
This post was
BOOKMARKED
divakarbio7 wrote:
If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?

A. 10
B. 12
C. 14
D. 16
E. 18

This question is really asking us to determine the number of 3's "hiding" in the prime factorization of p.

p = (1)(2)(3)(4)(5)(6)(7)(8)(9) . . . (27)(28)(29)(30)
= (1)(2)(3)(4)(5)(2)(3)(7)(8)(3)(3)(10)(11)(3)(4)(13)(14)(3)(5)(16)(17)(3)(3)(2)(19)(20)(3)(7)(22)(23)(3)(8)(25)(26). . . (3)(3)(3)(28)(29)(3)(10)
= (3)^14(other non-3 stuff)

[Reveal] Spoiler:
C

RELATED VIDEO

_________________

Brent Hanneson – Founder of gmatprepnow.com

Kudos [?]: 2743 [0], given: 364

Manager
Joined: 20 Jan 2017
Posts: 63

Kudos [?]: 8 [0], given: 15

Location: United States (NY)
Schools: CBS '20 (A)
GMAT 1: 750 Q48 V44
GMAT 2: 610 Q34 V41
GPA: 3.92
Re: If p is the product of the integers from 1 to 30, inclusive [#permalink]

### Show Tags

24 Jan 2017, 18:52
1) We need to calculate the number of prime factors 3 in the product of all the integers from 1 to 30
2) 30/3=10; 30/9=3; 30/27=1; 10+3+1=14

Kudos [?]: 8 [0], given: 15

Manager
Joined: 03 Jan 2017
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Re: If p is the product of the integers from 1 to 30, inclusive [#permalink]

### Show Tags

25 Mar 2017, 09:20
30/3, 30/9, 30/27
10+3+1=14

Kudos [?]: 9 [0], given: 4

Senior Manager
Joined: 04 Sep 2015
Posts: 479

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Location: India
WE: Information Technology (Computer Software)
Re: If p is the product of the integers from 1 to 30, inclusive [#permalink]

### Show Tags

25 Mar 2017, 23:35
If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?

A. 10
B. 12
C. 14
D. 16
E. 18

product of 1 to 30 includes the multiples of 3 and the number of 3 appear ,count them and add

3-1
6-1
9-2
12-1
15-1
18-2
21-1
24-1
27-3
30-1
=14

Kudos [?]: 81 [0], given: 18

Re: If p is the product of the integers from 1 to 30, inclusive   [#permalink] 25 Mar 2017, 23:35
Display posts from previous: Sort by
user
Given the text, the content structure is as follows:

The passage starts with a timestamp and a reference to a specific date, followed by a section header indicating "Decision(s) Day!" with links to chat rooms for various business schools. It then transitions into a section about GMAT Club Daily Prep, explaining the service and its features. The main content of the passage is a math problem related to factorials and prime factors.

TYPE: SINGLE_QUESTION_WITH_ANSWER

The passage contains a single mathematical question: "If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?" The answer to this question is provided in the discussion, with the correct answer being 14.