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# If p, q and r are integers, and pq + r

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Manager
Joined: 19 Aug 2016
Posts: 150
Location: India
GMAT 1: 640 Q47 V31
GPA: 3.82
If p, q and r are integers, and pq + r  [#permalink]

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12 Nov 2017, 10:35
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Difficulty:

95% (hard)

Question Stats:

52% (02:45) correct 48% (02:35) wrong based on 83 sessions

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If p, q and r are integers, and pq + r is an odd integer, is p an even integer?

(1) pq + pr is an even integer.
(2) p + qr is an odd integer.

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If p, q and r are integers, and pq + r  [#permalink]

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12 Nov 2017, 11:08
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1
ashikaverma13 wrote:
If p, q and r are integers, and pq + r is an odd integer, is p an even integer?

(1) pq + pr is an even integer.
(2) p + qr is an odd integer.

Given $$pq+r=Odd$$

Statement 1: $$pq+pr=Even$$, add $$r$$ to both sides

$$pq+r+pr=Even+r => Odd+pr=Even+r$$

or $$pr-r=Even-Odd => r(p-1)=Odd$$

$$1$$ is odd so $$p$$ has to be even because if $$p$$ is odd then $$p-1$$ is even and $$r(p-1)$$ will be even which is not possible. Sufficient

Statement 2: $$p+qr=Odd$$

so if $$p=even$$ & $$r=Odd$$ or $$p=odd$$, $$q=odd$$ and $$r=even$$. So no unique solution. Insufficient

Option A

-------------------------------------------

Another method. Plug in some values and test
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Manager
Joined: 03 Mar 2018
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Re: If p, q and r are integers, and pq + r  [#permalink]

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20 Mar 2018, 10:01
chetan2u
Even Though i got the answer right, I took more than 2 minutes here. Is there any way to solve this question within a minute or two. I arrived at the correct answer by assuming 2 cases for each statement. Like
pq+r = odd, So case 1 pq=odd; r= even
Case 2 pq=even; r=odd
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Re: If p, q and r are integers, and pq + r  [#permalink]

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20 Mar 2018, 17:38
3
1
itisSheldon wrote:
chetan2u
Even Though i got the answer right, I took more than 2 minutes here. Is there any way to solve this question within a minute or two. I arrived at the correct answer by assuming 2 cases for each statement. Like
pq+r = odd, So case 1 pq=odd; r= even
Case 2 pq=even; r=odd

Hi..
pq is common in statement I and the main statement..
So try and remove it to get a fresh equation..
pq+pr is EVEN and pq+r is odd..
Subtract them so pq+pr-(pq+r)=EVEN-ODD=ODD..
So pq+pr-pq-r=pr-r=(p-1)r is ODD
So both p-1 and r are ODD..
If p-1 is odd, p will be even
Suff
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Re: If p, q and r are integers, and pq + r  [#permalink]

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26 Mar 2018, 03:51

Solution

We are given:
• $$p$$, $$q$$, and $$r$$ are $$3$$ integers.
• $$(p*q + r)$$ is an odd integer.

$$(p*q + r)$$ can be odd for different cases, let us see:

Statement-1$$(p * q + p * r)$$ is an even integer.”

• We can conclude $$p * (q + r)$$ is an even integer.
• $$p * (q + r)$$ is an even integer for different cases among the above table.
1. $$p$$= Even, $$q$$= Even, $$r$$= Odd
2. $$p$$= Even, $$q$$= Odd, $$r$$= Odd

In both the cases, $$p$$ is an even integer.
Hence, Statement 1 alone is sufficient to answer the question.

Statement-2$$(p + q * r)$$ is an odd integer”.

• $$p + (q * r)$$ is an odd integer for different cases among the above table.
1. $$p$$ = Odd, $$q$$= Even, $$r$$=Odd
2. $$p$$ = Odd, $$q$$= odd, $$r$$=Even
3. $$p$$ = Even, $$q$$= Odd, $$r$$=Odd

$$p$$ can be even or odd.
Hence, Statement 2 alone is not sufficient to answer the question.

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Re: If p, q and r are integers, and pq + r &nbs [#permalink] 26 Mar 2018, 03:51
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