Thread: Positive Integers x & y

1. Positive Integers x & y

If positive integers x and y are NOT both odd, which of the following must be even?

A. xy
B. x + y
C. x - y
D. x + y - 1
E. 2(x + y) - 1

My Effort:

I decided to experiment by letting x be 3 and y be 2.
Doing this quickly revealed the fact that choice A and D both
yield an even number. The book's answer is A.

Question:

Why is choice D not the answer?

2. Re: Positive Integers x & y Originally Posted by harpazo If positive integers x and y are NOT both odd, which of the following must be even?

A. xy
B. x + y
C. x - y
D. x + y - 1
E. 2(x + y) - 1

My Effort:

I decided to experiment by letting x be 3 and y be 2.
Doing this quickly revealed the fact that choice A and D both
yield an even number. The book's answer is A.

Question:

Why is choice D not the answer?
What if $x~\&~y$ are both even then $x+y-1$ is ?

3. Re: Positive Integers x & y Originally Posted by harpazo Why is choice D not the answer?
Because the question talks about all possible pairs of integers, not just 2 and 3.

4. Re: Positive Integers x & y

A correct way to approach the question is, for each point A, B, C, D and E to consider the three permitted cases:
1. $x$ and $y$ are both even, that is $x=2a$, $y=2b$;
2. $x$ is even and $y$ is odd, that is $x=2a$, $y=2b-1$; and
3. $x$ is odd and $y$ is even, that is $x=2a-1$, $y=2b$.

Results that are even will have a factor of 2.

You might see that 2. and 3. here are essentially the same, so you only really need to do one of them. And you might be able to get by setting only $x=2a$ and considering different cases for $y$ where necessary. But essentially you will be doing the above with or without some shortcuts.

5. Re: Positive Integers x & y Originally Posted by harpazo If positive integers x and y are NOT both odd, which of the following must be even?
A. xy______B. x + y______C. x - y______D. x + y - 1______E. 2(x + y) - 1
Why is choice D not the answer?
To harpazo, I cannot understand how this can be so mysterious.
Learn this:
1. The sum of two even integers is even
2. The sum of two odd integers is even.
3. The sum of an even integer & an odd integer is odd.
4. If $n$ is an odd integer then $n-1$ is even.
5. If $n$ is an even integer then $n-1$ is odd.
If you learn these then practice applying them to this question,

6. Re: Positive Integers x & y Originally Posted by Plato What if $x~\&~y$ are both even then $x+y-1$ is ?
Let x = 6 and y = 4

x + y - 1

6 + 4 - 1

10 - 1 = 9 is an odd number.

Meaning?

7. Re: Positive Integers x & y Originally Posted by Archie Because the question talks about all possible pairs of integers, not just 2 and 3.
I can see that every word in word problems is important.

8. Re: Positive Integers x & y Originally Posted by Plato To harpazo, I cannot understand how this can be so mysterious.
Learn this:
1. The sum of two even integers is even
2. The sum of two odd integers is even.
3. The sum of an even integer & an odd integer is odd.
4. If $n$ is an odd integer then $n-1$ is even.
5. If $n$ is an even integer then $n-1$ is odd.
If you learn these then practice applying them to this question,
Good information.