Check Your Skills Answer Key

1. Odd

   You have an odd multiplied by an odd, which always results in an odd. Then add an even to the odd, which also results in an odd.

2. Odd

   At least one of the numbers multiplied together is even, meaning the product will be even. When you add an odd to that even, you get an odd.

3. Even

   Because integers go back and forth between evens and odds, the sum of any four consecutive integers can be expressed as Even + Even + Odd + Odd. Taking these one by one, start with Even + Even = Even. Then add an Odd to that Even, resulting in an Odd. Finally, add another Odd to that Odd, resulting in an Even.

4. No

   The product of two odd integers is always odd. Any multiple of two is even, and as the chart showed, an odd divided by an even cannot be an integer.

5. Yes

   Prime numbers only have two factors: 1 and themselves. So if the difference between the factors of a prime number is 1, its factors must be 1 and 2. This means x = 2. By the same logic, y must be equal to 3 (3 − 1 = 2). The product of 2 and 3 is 6, so xy is even.

6. (B) and (C)

   If x/y is even, then either x and y are both even, or x is even and y is odd. Make a chart:

   

   The question stem stipulates that x/y is even. This is only possible in the first two scenarios. In both of those situations, xy is even. This means that choice (A) is untrue, but choice (B) is true. While x + y can be either even or odd, that means that it could be odd, so choice (C) also works.

7. (C)

   Indeterminable. Once again, make a chart.

   The more restrictive constraints are the odd sums, which require either odd + even or even + odd. Thus, x and y have opposite status, as do y and z. There are only two such cases, as shown here:

   Scenario	x	y	z	xyz	x + z	y + z
   1	E	E	O	E	O	O
   2	O	O	E	E	O	O

   Therefore, xyz is even in either case, since there is always at least one even term in the product. As you can see, z is even in one case and odd in the other.