3. Given ‘x’ is an integer and y, z are two consecutive integers. Which of the following results an odd number?
A.xy
B.xy + z
C.
D.
Here's how to do these... use a 0 in place of an even number, and a 1 in place of an odd. With consecutive numbers, one is even and the other odd.
Substitute the 0's and 1's. If the answer is 0 or an even number, the result is even. If the answer is 1 or an odd number, the result is odd.
Since y and z are consecutive, one is odd and the other is even. However, we don't know which is which, and we know nothing about the parity of x.
A) What if x is even and y is not the odd one of y and z?
B) What if x is even, y is odd, and z is even?
C) Since only one of y and z can be even, then their sum must be odd, and then division by 2 gives you a fraction, because of the remainder.
D) The sum again has to be odd. It's fairly easy to show that the square then must also be odd. In particular, the square is not a multiple of 4.
Unless I'm missing something (always possible), none of these answers is correct.