If P is a set of integers and 3 is in P, is every positive multiple of 3 in P?

[BTGmoderatorDC]

If P is a set of integers and 3 is in P, is every positive multiple of 3 in P?

(1) For any integer in P, the sum of 3 and that integer is also in P.

(2) For any integer in P, that integer minus 3 is also in P.



OA A

Source: Princeton Review

[Jay@ManhattanReview]

If P is a set of integers and 3 is in P, is every positive multiple of 3 in P?

(1) For any integer in P, the sum of 3 and that integer is also in P.

(2) For any integer in P, that integer minus 3 is also in P.

OA A

Source: Princeton Review

Great question...

Let's understand the question.

The question asks whether set P is {..., 3, .., 6, ... 9, .., 12, ..15, ... ∞}

Note that set P can have many integers (Negative, 0, non-multiples of 3), but the question asks whether the set contains all positive multiples of 3, i.e., does the set contain 3, 6, 9, 12, 15, ....∞?

Let's take each statement one by one.

(1) For any integer in P, the sum of 3 and that integer is also in P.

The statement means that if an integer, say x, is in the set, then x + 3 is also in the set.

Since 3 is in the set, we have 3 + 3 = 6 in the set. This follows that 6 + 3 = 9 is in the set. Again, this follows that 9 + 3 = 12 is in the set. So, we can conclude that 3, 6, 9, 12, 15, ....∞ are in the set. The answer is yes. Sufficient.

Note: This does not mean that integers other than multiples of 3 may not be in the set; they maybe but we are not concerned in them.

(2) For any integer in P, that integer minus 3 is also in P.

The statement means that if an integer, say x is in the set, then x + 3 is there in the set.

Since 3 is there in the set, we have 3 – 3 = 0 in the set. This follows that 0 – 3 = –3 is in the set. Again, this follows that –3 – 3 = –6 is in the set. So, we can conclude that 3, 0, –3, –6, –9, ...∞ are in the set.

Note that there may or may not be all positive multiples of 3 in the set, but we do not know whether they are. So, we cannot call this statement SUFFICIENT on the basis that "No, set P does not contain all positive multiples of 3."

This statement is insufficient.

Another important aspect is that the answers from each statement cannot contradict. Since we know that statement 1 is sufficient on the basis that "Yes, set P contains all positive multiples of 3," if statement 2 is also sufficient then it can be sufficient only in one condition: YES, and not NO.

The correct answer: A

Hope this helps!

-Jay
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