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# If P is an odd integer and (P^2 + Q * R) is an even intege

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If P is an odd integer and (P^2 + Q * R) is an even intege  [#permalink]

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30 Jun 2017, 06:06
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Difficulty:

95% (hard)

Question Stats:

22% (01:19) correct 78% (01:08) wrong based on 164 sessions

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If P is an odd integer and (P^2 + Q * R) is an even integer, then which of the following must be true?

A. either Q or R is an odd integer
B. either Q or R is an even integer
C. both Q and R are odd integer
D. both Q and R are even integers
E. nothing can be concluded

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Re: If P is an odd integer and (P^2 + Q * R) is an even intege  [#permalink]

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30 Jun 2017, 06:21
1
mkrishnabdrr wrote:
If P is an odd integer and (P^2 + Q * R) is an even integer, then which of the following must be true?

A. either Q or R is an odd integer
B. either Q or R is an even integer
C. both Q and R are odd integer
D. both Q and R are even integers
E. nothing can be concluded

P^2 + QR = even

odd^2 + QR = even

odd + QR = even

QR = even - odd = odd.

If Q and R are integers, then both must be odd but we don't know that. So, they might not be integers at all (say $$Q=R=\sqrt{3}$$), which makes E the answer.

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Re: If P is an odd integer and (P^2 + Q * R) is an even intege  [#permalink]

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30 Jun 2017, 07:06
mkrishnabdrr wrote:
If P is an odd integer and (P^2 + Q * R) is an even integer, then which of the following must be true?

A. either Q or R is an odd integer
B. either Q or R is an even integer
C. both Q and R are odd integer
D. both Q and R are even integers
E. nothing can be concluded

We have 3 variable - P , Q and R

Given (P^2 + Q * R) = Even and P^2 = Odd

So, P = Odd

So, We can conclude that Q*R = Even, because Even*Even = Even & Odd*Even = Even

So, Q = Odd/Even & R = Odd/Even

Thus, we can not definitely conclude any value for Q or R, answer will be (E)
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Re: If P is an odd integer and (P^2 + Q * R) is an even intege  [#permalink]

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09 Aug 2017, 06:53
mkrishnabdrr wrote:
If P is an odd integer and (P^2 + Q * R) is an even integer, then which of the following must be true?

A. either Q or R is an odd integer
B. either Q or R is an even integer
C. both Q and R are odd integer
D. both Q and R are even integers
E. nothing can be concluded

P is an odd integer
(P^2 + Q * R) is an even integer

P^2 is also an odd integer... So, for (P^2 + Q * R) to be an even integer, Q*R must be odd integer.
Now Q*R can be odd in two cases.

1st : Both are integers : In this case Q and R both are odd integers.

2nd : One of them is integer and other is fraction : In this case the first integer may be odd or even and second integer is a fraction.

So, Nothing can be concluded about Q & R in terms of being odd or even or integral form.

This is a very tricky question. People, including me, get caught in the trap and mark the wrong answer as C. Be very careful while reading such questions... A simple mistake will cost your marks. Finally don't forget to appreciate.
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Re: If P is an odd integer and (P^2 + Q * R) is an even intege  [#permalink]

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26 Jun 2018, 17:17
mkrishnabdrr wrote:
If P is an odd integer and (P^2 + Q * R) is an even integer, then which of the following must be true?

A. either Q or R is an odd integer
B. either Q or R is an even integer
C. both Q and R are odd integer
D. both Q and R are even integers
E. nothing can be concluded

Solution:

Since P is odd, P^2 is also odd.In order for P^2 + Q * R to be even, Q * R must be odd, since odd + odd = even. If we knew that both Q and R were integers, then necessarily both Q and R would have to be odd. However, Q could be 2 and R could be 1/2, in which case the product Q * R = 2 x 1/2 = 1 is still an odd integer. Therefore, we cannot conclude anything about the parity of Q and R. Thus, the answer is E.

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Re: If P is an odd integer and (P^2 + Q * R) is an even intege  [#permalink]

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09 Oct 2018, 22:35
Hi All,

We're told that If P is an ODD integer and (P^2 + Q * R) is an EVEN integer. We're asked which of the following MUST be true. This question can be solved with a mix of Number Properties and TESTing VALUES.

To start, since P is ODD, we know that P^2 will also be ODD (since ODD^2 = ODD).
For (ODD + Q * R) to be EVEN, we know that (Q * R) must also be ODD. However we do NOT know whether Q and R are integers or not...

IF Q=1 and R=1, then all 3 variables are ODD.
IF Q=2 ad R = 1/2, then the 3 variables include one ODD, one EVEN and one non-integer.

Thus, none of the first 4 answers is always going to be true.

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Re: If P is an odd integer and (P^2 + Q * R) is an even intege   [#permalink] 09 Oct 2018, 22:35
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