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Question
In the given question, there are two statements marked I and II. Decide which of the statements are sufficient to answer the question. Choose your answer from the given alternative.
If p, q, and'are integers, is pq + r even ?
(I) p + r is even.
(II) q + r is odd.
In the given question, there are two statements marked I and II. Decide which of the statements are sufficient to answer the question. Choose your answer from the given alternative.
(I) p + r is even.
(II) q + r is odd.
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Solution
The correct option is D Statement I and II both are not sufficient.
The yes/no question asks whether pq + r is even. What would need to be true in order for the answer to be yes? Either both pq and r need to be even or both pq and r need to be odd.(1) INSUFFICIENT: You are told that p + r is even. To stay organized, test all the cases that make the statement true. Both p and r are even, or both p and r are odd. For each of those scenarios, q could be odd or even. Set up a table to keep track of all of these possibilities:Scenario p q r pq + r1 Odd Odd Odd O×O+O=E2 Odd Even Odd O×E+O=O3 Even Odd Even E×O+E=E4 Even Even Even E×E+E=ESince pq + r could be odd or even, statement (1) is not sufficient. Note that you can stop as soon as you have found contradictory cases (one odd and one even); above, for example, you could have stopped after Scenario 2.(2) INSUFFICIENT: As in statement (1), you can organize the information from statement (2) with a table. Either q is even and r is odd or q is odd and r is even, and p can be odd or even:Scenario p q r pq + r5 Odd Even Odd O×E+O=O6 Even Even Odd E×E+O=O7 Odd Odd Even O×O+E=O8 Even Odd Even E×O+E=E(1) AND (2) INSUFFICIENT: Notice that Scenarios 2 and 5 are identical, as are Scenarios 3 and 8. Therefore, both sets of scenarios meet the criteria laid forth in statements (1) and (2), but they yield opposite answers to the question:Scenario p q r pq + r2 & 5 Odd Even Odd O×E+O=O3 & 8 Even Odd Even E×O+E=E
The yes/no question asks whether pq + r is even. What would need to be true in order for the answer to be yes? Either both pq and r need to be even or both pq and r need to be odd.
(1) INSUFFICIENT: You are told that p + r is even. To stay organized, test all the cases that make the statement true. Both p and r are even, or both p and r are odd. For each of those scenarios, q could be odd or even. Set up a table to keep track of all of these possibilities:
Scenario p q r pq + r
1 Odd Odd Odd O×O+O=E
2 Odd Even Odd O×E+O=O
3 Even Odd Even E×O+E=E
4 Even Even Even E×E+E=E
Since pq + r could be odd or even, statement (1) is not sufficient. Note that you can stop as soon as you have found contradictory cases (one odd and one even); above, for example, you could have stopped after Scenario 2.
(2) INSUFFICIENT: As in statement (1), you can organize the information from statement (2) with a table. Either q is even and r is odd or q is odd and r is even, and p can be odd or even:
Scenario p q r pq + r
5 Odd Even Odd O×E+O=O
6 Even Even Odd E×E+O=O
7 Odd Odd Even O×O+E=O
8 Even Odd Even E×O+E=E
(1) AND (2) INSUFFICIENT: Notice that Scenarios 2 and 5 are identical, as are Scenarios 3 and 8. Therefore, both sets of scenarios meet the criteria laid forth in statements (1) and (2), but they yield opposite answers to the question:
Scenario p q r pq + r
2 & 5 Odd Even Odd O×E+O=O
3 & 8 Even Odd Even E×O+E=E
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