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

For pq+r to be even, pq = even and r = even OR pq = odd and r = odd.

From (1)
p + r is even so p is even and r is even or p is odd and r is odd

but no info about q, if q is even and p and r are odd, then pq+r is odd, otherwise even



From (2) , q + r is odd, so q is even and r is odd OR q is odd and r is even

So if p is even, then pq -> even + r (odd) = odd or even + r (even) = even

Not sufficient

Combining both (1) and (2), if p and r ar even, pq + r is even

if p and r are odd then, pq + r is odd (as q is even then)

Hence we don't have definitive answer,and the answer is E.

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

(1) p + r is even.
(2) q + r is odd.

Manhattan tells me I should make the table which works fine. I tried doing it without the table and that worked too. However, without the table I was less convinced and more confused because in your head it gets jumbled up. So is there another foolproof way of doing these? Or do I have to stick with the Manhattan table?

The Manhattan table works fine, another method is using real numbers .

(1) p + r is even. \(3+1 = even\), so is \(3q+1\) even? depends on q : not Sufficient
(2) q + r is odd. \(2+1=odd\), so is \(p2+1\) even? depends on p : not Sufficient

(1)+(2) p + r is even AND q + r is odd
Example 1: \(3+1=even\)--\(2+1 = odd\)
\(2*3+1=odd\)
Example 2:\(2+2=even\)--\(3+2=odd\)
\(2*3+2=even\)
Not Sufficient

From F.S 1, assume p=r=0, thus, we get a YES for the question stem. Now assume p=1, r=1,q = 2 we get a NO. Insufficient.

From F.S 2, assume q=0,r=1, we get a NO for the question stem.Now assume r=2,q=1 ,p=2, we get a YES. Insufficient.

Taking both together, we have p=0,r=0,q=1, and a YES. Again taking, r=1,p=1,q=0, a NO. Insufficient.

What might help you in selecting good numbers is the fact that from the F.S 1,either both p,r are even or both are odd. Similarly, from F.S 2, q and r are odd/even or even/odd.

E.

Odds and Evens, ok

Statement 1

Clearly Insufficient

Statement 2

Same here

Statements 1 and 2 combined

p+r = even
q+r = odd

p-q = odd

Then p must be even and q odd or the other way around

If p is even then pq will be even and 'r' will be even = All even= Answer is YES
if q is even then pq will again be even and 'r' will be odd= All odd = Answer is NO

Hence E is your best choice

Cheers!
J

chetan2u niks18 pushpitkc pikolo2510 VeritasPrepKarishma

I was not able to combine statements correctly after individually assessing them.




This means either p or r is even or BOTH are even.
Since we can not have a definite ans about p,
first term (ie p*q) can be even or odd. Since
we can not definitely know that r is even or odd, final sum
of p*q and r can be either even or odd


This means either q or r is odd.
Hence first term (ie p*q) can be even or odd. Since
we can not definitely know that r is even or odd, final sum
of p*q and r can be either even or odd

How do we analyze after combing both statements and decide between C/E

1) P+R IS EVEN
both P and R is even or both odd..
2) q+r is odd
one is odd and other is even, so q is opposite of r

combined..
pq+r
A) let both p and r be even then q will be opposite of r so odd
pq+r = E*O+E=E+E=Even
B) let both p and r be ODD then q will be opposite of r so even
pq+r = E*O+O=E+O=Odd

so both E and O possible
insuff

Hey Arpit,

Make a table of P , Q and R

when you combine, the possibilities are

P-Q-R
1) E-O-E
2) O-E-O

Now check them for pq+r. I think making a quick table will help you clear the confusions

answer will be E

(1) insufficient
Ex+e result is even
Ox+o result is odd if x is even, even if x is odd
Insufficient

(2) insufficient
Xe+o odd if x is even, odd if x is odd
Xo+e even if x is even, odd if x is odd

(1) and (2) are insufficient
Eo+e even
Oe+o odd

Thank you for the responses. I understand how neither S1 and S2 separately are sufficient and I understand theoretically how E could be the right answer. But when we combine the two statements, if we apply the GMAT principle that the statements cannot conflict each other, how is our answer not C?

S1: P + R = Even
P = Odd, then R = Odd
P = Even, then R = Even

S2: Q - R = Odd
Q = Even, then R = Odd
Q = Odd, then R = Even

Since R is included in both statements, I'll propose our answers by flexing the R variable, which in turn drives our other variables:

1) Looking at S1, if R is Odd, then for S1 to hold true, P must be Odd, and in order for S2 not to conflict with S1 (i.e. R is Odd), then Q must be Even
Applying to the Question stem then: Odd(Odd) + Odd = Even
2) Looking at S1, If R is Even, then for S1 to hold true, P must be Even, and in order for S2 not to conflict with S1 (i.e. R is Even), then Q must be Odd
Applying to the Question stem then: Even(Odd) + Even = Even

Am I misinterpreting the GMAT principle of how the statements cannot conflict each other? Thank you in advance for the help.

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