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# M03-22

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Math Expert
Joined: 02 Sep 2009
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16 Sep 2014, 00:20
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Difficulty:

35% (medium)

Question Stats:

62% (01:05) correct 38% (01:00) wrong based on 231 sessions

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Is the product of three integers $$p$$, $$q$$, and $$r$$ even?

(1) $$(p - 1)(r + 1)$$ is odd

(2) $$(q - r)^2$$ is odd

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16 Sep 2014, 00:20
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Official Solution:

For a product of 3 integers to be even, at least one integer must be even. For example $$1 * 3 * 5 = 15$$, odd vs. $$5 * 7 * 2 = 70$$, even.

Statement (1) by itself is sufficient. For $$(p - 1)(r + 1)$$ to be odd, both $$p - 1$$ and $$r + 1$$ must be odd. Therefore, both $$p$$ and $$r$$ are even.

Statement (2) by itself is sufficient. The product of two integers is odd only if both integers are odd, therefore the result of $$q - r$$ is odd. For a difference of two integers to be odd, one of the two must be even. One even integer is sufficient to make the product $$pqr$$ even.

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29 Dec 2014, 02:25
Bunuel But what about if q is 0 for the first one or if q or r is 0 for the second one?
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29 May 2015, 05:08
zero is even number
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29 May 2015, 07:02
Statement 1: (p-1)(r+1) = (pr+p-r-1). This will be odd only (pr+p-r) is even. (pr+p-r) is even only when p and r are even. Since p and r are even. Product of p, q and r is always even. (Sufficient)

Statement 2: (q-r)^2 is odd when (q-r) is odd. (q-r) will be odd when either q or r is even and the other one is odd. Since q or r is even, product of p, q and r will always be even. (Sufficient)

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23 Oct 2015, 11:46
I think this is a high-quality question and I don't agree with the explanation. For option 2, is it not possible that q is odd and r is 0, which makes (q-r) * (q-r) as odd and hence the statement is not sufficient because it will depend on the value of P.

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24 Oct 2015, 02:28
mohitmaheshwari20 wrote:
I think this is a high-quality question and I don't agree with the explanation. For option 2, is it not possible that q is odd and r is 0, which makes (q-r) * (q-r) as odd and hence the statement is not sufficient because it will depend on the value of P.

You should brush-up fundamentals before attempting questions: 0 is an even integer.
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24 Feb 2016, 19:01
I think this is a poor-quality question and I don't agree with the explanation. No you don't know if Q is 0 or not for 1) or if P is 0 in 2)
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24 Feb 2016, 19:45
Binglai wrote:
I think this is a poor-quality question and I don't agree with the explanation. No you don't know if Q is 0 or not for 1) or if P is 0 in 2)

Hi,
the Q is perfectly fine..
you will have to brush up a bit of basics on number, since 0 is an even number and is neither positive or negative..

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11 May 2016, 16:55
I think this is a poor-quality question and I don't agree with the explanation. What if one of p, r, q was zero?
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11 May 2016, 17:24
mengsy wrote:
I think this is a poor-quality question and I don't agree with the explanation. What if one of p, r, q was zero?

You should brush-up fundamentals before attempting questions: 0 is an even integer.
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03 Aug 2016, 07:26
Bunuel wrote:
mengsy wrote:
I think this is a poor-quality question and I don't agree with the explanation. What if one of p, r, q was zero?

You should brush-up fundamentals before attempting questions: 0 is an even integer.

I think we must keep our politeness whenever possible. The questions are repeated over and over. The simple fact of not getting it right worth analyzing it and putting it in you error log. Next time this mistake won't happen.
>> !!!

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22 Aug 2017, 16:01
I think this is a high-quality question and I agree with explanation.
Re M03-22   [#permalink] 22 Aug 2017, 16:01
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# M03-22

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