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Re: For each positive integer n, p(n) is defined to be the product of the [#permalink]
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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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Re: For each positive integer n, p(n) is defined to be the product of.. [#permalink]
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]
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.
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For each positive integer n, p(n) is defined to be the product of.. [#permalink]
Expert Reply
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!
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Re: For each positive integer n, p(n) is defined to be the product of.. [#permalink]
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.
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Re: For each positive integer n, p(n) is defined to be the product of.. [#permalink]
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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,
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Re: For each positive integer n, p(n) is defined to be the product of.. [#permalink]
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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]
Great question - need to plug in values to solve and find 'must be true'
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