ziyuenlau wrote:x, y, and z are positive integers and x < y < z. If x+y+z is an even number less than 15, is x equal to 2?

1) The product of any two numbers of x, y, and z is an even number.
2) One of x, y, and z is twice the one of the other numbers.

Source : Math Revolution
Official Answer : A

First, if the sum of three numbers is EVEN, we only have two possibilities. 1) EVEN/EVEN/EVEN and 2)ODD/ODD/EVEN. Let's keep that in mind as we evaluate the statements.

1) If the product of any two numbers is EVEN, that means that only the first scenario: EVEN/EVEN/EVEN will work. So we have three EVEN numbers whose sum is less than 15. Let's see if we can generate multiple scenarios. Case 1: x = 2, y = 4, z = 6. Here YES, x = 2. Note, nothing else will work. If x = 4, then the smallest values for y and z are 6 and 8 respectively. But 4 + 6 + 8 is not less than 15, so these numbers are not eligible to test. Thus, x must be 2, and statement 1 alone is sufficient.

2) Let's test cases. Note that we can reuse Case 1 from the previous statement: x = 2, y = 4, z = 6, as the value of y is twice the value of x. Here YES, x = 2.

But we could also try an instance of ODD/ODD/EVEN. x = 1, y = 2, z = 3. (Again, y is twice the value of x.) Now x is not 2, so we get a NO. Statement 2 alone is not sufficient to answer the question.

The answer is A

hazelnut01 wrote:x, y, and z are positive integers and x < y < z. If x+y+z is an even number less than 15, is x equal to 2?

1) The product of any two numbers of x, y, and z is an even number.
2) One of x, y, and z is twice the one of the other numbers.

We are given that x, y, and z are all positive integers, that x < y < z, and that x + y + z is an even number less than 15. We need to determine whether x = 2.

If the sum of three integers is even, then either 1) all integers are even or 2) one of them is even and the other two are odd.

Statement One Alone:

The product of any two numbers of x, y, and z is an even number.

Using the information in statement one, we know that at least two of the three numbers, x, y, and z, have to be even numbers. That is, either 1) all three numbers are even or 2) two of them are even and the other is odd. However, it can't be the latter case because in the stem analysis, we see that we cannot have two even integers and one odd integer. Thus all three numbers must be even.

Since the sum of x, y, and z is less than 15, the only even numbers that will work for x, y, and z, respectively, are 2, 4, and 6. Thus, x = 2. Statement one alone is sufficient to answer the question.

Statement Two Alone:

One of x, y, and z is twice the one of the other numbers.

The information in statement two is not sufficient to answer the question. For instance, x = 2, y = 4, and z = 6, or x = 1, y = 2, and z = 3.

Answer: A