Data Sufficiency

Jay@ManhattanReview wrote:

lheiannie07 wrote:Set A consists of 8 distinct prime numbers. If x is equal to the range of set A and y is equal to the median of set A, is the product xy even?

(1) The smallest integer in the set is 5.
(2) The largest integer in the set is 101

How can i determine the correct statement? Can experts explain? Thanks

OAA

You must know that 2 is the only even prime or all the prime numbers except 2 are odd.

So, Set A may have all the odd prime numbers or have one even (2) and seven prime numbers.

Case 1: Set A: {2, OP1, OP2, OP3, OP4, OP5, OP6, OP7}; OP: Odd Prime
OP1 means first prime, OP5 means fifth prime, etc.

x = Range = OP7 - 2 = Odd number
y = Median = (OP3 + OP4)/2 = Even/2 = Odd or Even
-- If OP3 = 7 and OP4 = 11, then y = (7 + 11)/2 = 18/2 = 9 (Odd);
-- If OP3 = 11 and OP4 = 13, then y = (11 + 13)/2 = 24/2 = 12 (Even)

So, xy can be even or odd.

So, there are two ways to get the product xy even.

-- If x is even, then xy is even. For x to be even, the set must not have 2 as a prime number. This way, x = Range = Odd - Odd = Even

Or,

-- If y is even, then xy is even. For y to be even, the 4th and the 5th odd numbers must be such that their sum divided by 2 gives an even number. For example, 11 and 13.

Case 2: Set A: {OP1, OP2, OP3, OP4, OP5, OP6, OP7, OP8}

x = Range = OP8 - OP7 = Even

We need not bother about the nature about y since x is even.

As discussed above that if 2 is not there as one of the primes, then the answer is Yes.

Statement 1: The smallest integer in the set is 5.

This implies that 2 is not there in the set as one of the prime numbers, thus x = range is even, and xy is even. Sufficient.

Statement 2: The largest integer in the set is 101.

Let's form a set once with the inclusion of 2 and once without 2.

Case 1: {2, 3, 5, 7, 11, 13, 17, 101} --> x = Range is odd, and y = Median = (7 + 11)/2 = 9 (odd), thus, xy is odd.
Case 2: {3, 5, 7, 11, 13, 17, 23, 101} --> x = Range is even, thus, xy is even.

No unique answer. Insufficient.

The correct answer: A

Hope this helps!

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-Jay
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