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# For each positive integer n, p(n) is defined to be the product of..

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Intern
Joined: 17 Jan 2016
Posts: 10
For each positive integer n, p(n) is defined to be the product of..  [#permalink]

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Updated on: 05 May 2016, 10:39
4
44
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Difficulty:

55% (hard)

Question Stats:

63% (01:51) correct 37% (01:53) wrong based on 559 sessions

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For each positive integer $$n$$, $$p(n)$$ is defined to be the product of the digits of $$n$$. For example, $$p(724) = 56$$ since $$7 * 2 * 4 =56$$.

Which of the following statements must be true?

I. $$p(10n) = p(n)$$

II. $$p(n+1) > p(n)$$

III. $$p(2n) = 2p(n)$$

--

A. None
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

Originally posted by broilerc on 05 May 2016, 10:16.
Last edited by Vyshak on 05 May 2016, 10:39, edited 1 time in total.
Formatted the question
##### Most Helpful Expert Reply
Target Test Prep Representative
Status: Head GMAT Instructor
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2815
Re: For each positive integer n, p(n) is defined to be the product of..  [#permalink]

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30 Aug 2016, 16:40
13
10
broilerc wrote:
For each positive integer $$n$$, $$p(n)$$ is defined to be the product of the digits of $$n$$. For example, $$p(724) = 56$$ since $$7 * 2 * 4 =56$$.

Which of the following statements must be true?

I. $$p(10n) = p(n)$$

II. $$p(n+1) > p(n)$$

III. $$p(2n) = 2p(n)$$

--

A. None
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

Let’s go through each statement given in the Roman numerals.

I. p(10n) = p(n)

This is not true. For example, if n = 12, p(12) = 2 since 1 x 2 = 2. However, 10n = 10(12) = 120 and p(120) = 0 since 1 x 2 x 0 = 0. Since p(120) ≠ p(12), p(10n) = p(n) is not a true statement.

II. p(n +1) > p(n)

This is not true. For example, if n = 19, then p(19) = 9 since 1 x 9 = 9. However, n + 1 = 19 + 1 = 20 and p(20) = 0 since 2 x 0 = 0. Since p(20) < p (19), p(n +1) > p(n) is not a true statement.

Since neither I nor II is true, it can’t be choices B, C, D or E. So the correct choice must be A. However, let’s show III is also not true.

III. p(2n) = 2p(n)

For example, if n = 15, then p(15) = 5 since 1 x 5 = 5 and 2p(15) = 2 x 5 = 10. However, 2n = 2 x 15 = 30 and p(30) = 0 since 3 x 0 = 0. Since p(30) ≠ 2p(15), p(2n) = 2p(n) is not a true statement.

Answer: A
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##### Most Helpful Community Reply
Marshall & McDonough Moderator
Joined: 13 Apr 2015
Posts: 1684
Location: India
Re: For each positive integer n, p(n) is defined to be the product of..  [#permalink]

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05 May 2016, 10:37
10
6
I) p(10n) = p(n)
p(10n) will always have units digit as 0 --> p(10n) = 0
p(n) can be any integer.
Not a must be true statement

II) p(n + 1) > p(n)
If n = 2; p(n +1) = 3 and p(n) = 2 --> p(n + 1) > p(n)
If n = 9; p(n + 1) = 0 and p(n) = 9 --> p(n + 1) < p(n)
Not a must be true statement

III) p(2n) = 2*p(n)
If n = 12; p(2n) = p(24) = 8 and 2*p(n) = 4
Not a must be true statement

Answer: A
##### General Discussion
Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4472
Re: For each positive integer n, p(n) is defined to be the product of the  [#permalink]

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17 Jun 2016, 16:47
1
tanad wrote:
For each positive integer n, p(n) is defined to be the product of the digits of n. For example, p(724) = 56, since 7 x 2 x 4 = 56. Which of the following statements must be true?

I. p(10n) = p(n)
II. p(n + 1) > p(n)
III. p(2n) = 2p(n)

A: None
B: I and II only
C: I and III only
D: II and III only
E: I, II and III

Dear tanad,
I'm happy to respond.

My friend, I gather that you are relatively new to GMAT Club. I will share with you an important piece of GC etiquette. Please do NOT start a brand new thread for a question that has already been posted. This particular question has already be posted here:
for-each-positive-integer-n-p-n-is-defined-to-be-the-product-of-217887.html
Always search before you start a separate post. You may find your question answered in that post, and if you don't, you can add your own question to that existing thread. Bunuel will merge this post into that thread.

Does all this make sense?
Mike
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Mike McGarry
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Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
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Re: For each positive integer n, p(n) is defined to be the product of..  [#permalink]

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29 Aug 2016, 14:08
Bunuel wrote:
tanad wrote:
For each positive integer n, p(n) is defined to be the product of the digits of n. For example, p(724) = 56, since 7 x 2 x 4 = 56. Which of the following statements must be true?

I. p(10n) = p(n)
II. p(n + 1) > p(n)
III. p(2n) = 2p(n)

A: None
B: I and II only
C: I and III only
D: II and III only
E: I, II and III

___________________________
Merging topics.

Hi Bunuel, can you please help to answer?
Read the response by Vyshak and still confused. Didn't see other responses so reaching out.

Thanks!
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Re: For each positive integer n, p(n) is defined to be the product of..  [#permalink]

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31 Aug 2016, 04:48
JeffTargetTestPrep wrote:
broilerc wrote:
For each positive integer $$n$$, $$p(n)$$ is defined to be the product of the digits of $$n$$. For example, $$p(724) = 56$$ since $$7 * 2 * 4 =56$$.

Which of the following statements must be true?

I. $$p(10n) = p(n)$$

II. $$p(n+1) > p(n)$$

III. $$p(2n) = 2p(n)$$

--

A. None
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

Let’s go through each statement given in the Roman numerals.

I. p(10n) = p(n)

This is not true. For example, if n = 12, p(12) = 2 since 1 x 2 = 2. However, 10n = 10(12) = 120 and p(120) = 0 since 1 x 2 x 0 = 0. Since p(120) ≠ p(12), p(10n) = p(n) is not a true statement.

II. p(n +1) > p(n)

This is not true. For example, if n = 19, then p(19) = 9 since 1 x 9 = 9. However, n + 1 = 19 + 1 = 20 and p(20) = 0 since 2 x 0 = 0. Since p(20) < p (19), p(n +1) > p(n) is not a true statement.

Since neither I nor II is true, it can’t be choices B, C, D or E. So the correct choice must be A. However, let’s show III is also not true.

III. p(2n) = 2p(n)

For example, if n = 15, then p(15) = 5 since 1 x 5 = 5 and 2p(15) = 2 x 5 = 10. However, 2n = 2 x 15 = 30 and p(30) = 0 since 3 x 0 = 0. Since p(30) ≠ 2p(15), p(2n) = 2p(n) is not a true statement.

Answer: A

Thanks Jeff! Clear now.
Btw, did you think to pick specifically those numbers to test? I see you tried to pick ones were you would end up with a 0 as a multiplier.
Target Test Prep Representative
Status: Head GMAT Instructor
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2815
For each positive integer n, p(n) is defined to be the product of..  [#permalink]

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31 Aug 2016, 04:54
18967mba wrote:
JeffTargetTestPrep wrote:
broilerc wrote:
For each positive integer $$n$$, $$p(n)$$ is defined to be the product of the digits of $$n$$. For example, $$p(724) = 56$$ since $$7 * 2 * 4 =56$$.

Which of the following statements must be true?

I. $$p(10n) = p(n)$$

II. $$p(n+1) > p(n)$$

III. $$p(2n) = 2p(n)$$

--

A. None
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

Let’s go through each statement given in the Roman numerals.

I. p(10n) = p(n)

This is not true. For example, if n = 12, p(12) = 2 since 1 x 2 = 2. However, 10n = 10(12) = 120 and p(120) = 0 since 1 x 2 x 0 = 0. Since p(120) ≠ p(12), p(10n) = p(n) is not a true statement.

II. p(n +1) > p(n)

This is not true. For example, if n = 19, then p(19) = 9 since 1 x 9 = 9. However, n + 1 = 19 + 1 = 20 and p(20) = 0 since 2 x 0 = 0. Since p(20) < p (19), p(n +1) > p(n) is not a true statement.

Since neither I nor II is true, it can’t be choices B, C, D or E. So the correct choice must be A. However, let’s show III is also not true.

III. p(2n) = 2p(n)

For example, if n = 15, then p(15) = 5 since 1 x 5 = 5 and 2p(15) = 2 x 5 = 10. However, 2n = 2 x 15 = 30 and p(30) = 0 since 3 x 0 = 0. Since p(30) ≠ 2p(15), p(2n) = 2p(n) is not a true statement.

Answer: A

Thanks Jeff! Clear now.
Btw, did you think to pick specifically those numbers to test? I see you tried to pick ones were you would end up with a 0 as a multiplier.

Yeah, so I thought it would be easiest to select numbers that had a zero as one of the digits. Glad I could help!
_________________

# Jeffrey Miller

Head of GMAT Instruction

Jeff@TargetTestPrep.com
122 Reviews

5-star rated online GMAT quant
self study course

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

Retired Moderator
Joined: 18 Sep 2014
Posts: 1093
Location: India
Re: For each positive integer n, p(n) is defined to be the product of..  [#permalink]

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21 Oct 2016, 11:22
broilerc wrote:
For each positive integer $$n$$, $$p(n)$$ is defined to be the product of the digits of $$n$$. For example, $$p(724) = 56$$ since $$7 * 2 * 4 =56$$.

Which of the following statements must be true?

I. $$p(10n) = p(n)$$

II. $$p(n+1) > p(n)$$

III. $$p(2n) = 2p(n)$$

--

A. None
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

let n=23
Consider III
P(2n)=p(46)=4*6=24
p(n)=2*3=6
24 is not equal to two times of 6.........so III is not must be true ruling out C, D and E options.

Come to I, any number multiplied by ten will have 0 as one of its integers and product will be zero. Ruling out B as well leaving only A as correct choice.
Intern
Joined: 08 Mar 2017
Posts: 1
Re: For each positive integer n, p(n) is defined to be the product of..  [#permalink]

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22 Jan 2018, 10:22
Hello,

They mentionned for any positive number n the multiplication works. so p(10n)=1*n without multiplying it by 0 because 0 is not considered positive.
What is the part I don't get?

Best,
G
Intern
Joined: 30 Nov 2017
Posts: 23
Re: For each positive integer n, p(n) is defined to be the product of..  [#permalink]

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15 Feb 2018, 07:27
It was easy to eliminate I and III, but I initially thought that II was correct.

Thankfully "II only" was not in the answer choices, which made me think again. Then I realized that if the last digit was 9, then II will be violated.
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Re: For each positive integer n, p(n) is defined to be the product of..  [#permalink]

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28 Feb 2019, 08:08
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Re: For each positive integer n, p(n) is defined to be the product of..   [#permalink] 28 Feb 2019, 08:08
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